Segment 01:
Obviously, this segment is far from perfect, but after being presented with an insane amount of random battle luck, I could not stand the thought of doing the segment over again. Thankfully, Cloud's and Barret's stats did not force me to do so either.

Segment 02:
Pretty much just lolo's strategy.

Segment 03:
Just running arond and text skipping.

Segment 04:
I remember trying different enemy encounters, and this sequence, which requires soft resetting, seemed to work best. The walk exploit (3:17) was discovered by BrutalAl and first used by GarlandTheGreat.

Segment 05:
This is the standard Air Buster strategy (for segmented runs that don't deem a Pre-emptive attack on Aps necessary). Also, one does not really need the Cover Materia.

Segment 06:
I need to keep at least 280 Gil to pay for what it takes to have Don Corneo choose Cloud.

Segment 07:
First of all, you most certainly want to be able to throw a Grenade at Aps before he gets his first attack as this will shorten the battle by quite a bit. Of course, that is rather rare to occur. Secondly, you must not get stuck in Wall Market, the Sewers or the Train Graveyard, which means you will spend a long time practicing how to properly run around. Finally, it is important to escape the two random battles as quickly as possible, which is also rather rare, and for good measure, you should by all means avoid a third random battle. This is the aggravating part, for, in case your movement is suboptimal, you will definitely trigger that dreadful third random encounter just shortly before exiting the Train Graveyard. Now, combining all these restrictions yields a not so jolly probability of making it through this segment. In fact, it proved to be so annoying that I decided to accept these horrible movement mistakes I made in the Sewers and even more so in the Train Graveyard.

Segment 08:
In this segment, movement was the bane of my existence. If one makes merely very slight running mistakes, one will most likely get a second random encounter on the way back to Sector 5. I don't recall exactly how often I had to redo this part, but I am pretty certain it took me more than one hundred attempts. In the end, I was only this one time able to arrive at Sector 5 without an additional random battle and, even though I made a lot of very ugly, awkward mistakes, I did not really care any longer for them. Granted, I possibly could have saved 10 seconds if I had redone this segment a few more hundred times, but that just seemed silly back then and I still don't regret these lost seconds all that much. Oh yeah, did I mention that you will also want to escape the random battle on the staircase as quickly as possible and have Reno refrain from using Electro-mag Rod as it may paralyze and has a longer attack animation than his other attack?

Segment 09:
I am nearly tempted to use the word impeccable to describe this segment. Yet, it contains running (almost exclusively), so you can guess what that means...

Segment 10:
This is definitely one of the hardest segments in the run. First of all, it is rather long, but much more importantly, it comes down almost entirely to luck, elevator luck. Obviously, taking the elevator CAN be much faster than taking the stairs, however, whether that will be the case is random. Here is what you need to know about the elevator: You start at floor 6 and need to reach floor 59 (you can't go higher than that). The elevator will ascend by 7 to 15 levels; the chance of going up by any number of levels from 7 to 13 is each 1/8 and the chance of going up by 14 or 15 levels is each 1/16. Depending on what floor the elevator comes to a temporary halt at, you will either have to fight an unescapable battle or see a Shinra employee, which, of course, is the preferable option. The best outcome is having to fight only one random battle and running into the Shinra employee 3 times, which will happen just about never outside of a TAS.
I did this segment probably around 800 times to end up with the outcome you can find in the video. In case you get a nice elevator sequence, you still need to manage to get through around 9 minutes of pointless conversations, quiz time and running around. I decided to be utterly risk-averse on floor 60 as a very promising earlier attempt ended just there... On floor 62, a keyword needs to be "guessed" on the first try to get an Elemental materia. As you probably already know, all you have to do is read that one bookshelf and note what it says:
2 Economic... - BEST
1 Results... - KING
1 Breakdown... - BOMB
10 New... - MAKO
Floor 65 is where I make a lot of running mistakes. Still, they hardly amount to 10 seconds and considering how lucky I got with the elevator, I could not care less about them. Also, floor 65 has a trick, which a certain Speedrun Guide by a certain someone does not mention (but ff7.fr does...): If you put the first Parts of Midgar into the lowermost position in the central room, every chest (apart from the one holding the Keycard 66 for obvious reasons) you try to open will be unlocked. Granted, that does not help to save time, however, if you insert the first Pieces of Midgar into any other position, only one treasure chest will be unlocked and you have to determine which by where you put the last Parts of Midgar, which means you would have to take even more notes/memorize more things.

Segment 11:
The three Sample:H0512-opt enemies in the boss battle were the cause of some headaches. Basically, they may not attack even once or quite a lot. In addition thereto, Sample:H0512 will use the poison inflicting Shady Breath at the beginning of the battle but, if not all characters end up posioned, he may use it again.Obviously, seeing the Sample:H0512-opts attack (frequently) and Sample:H0512 use Shady Breath more than once equals a time loss. As this is a speedrun (just in case you forgot), a time loss equals a No-Go and thus, this battle equals a lot of resets. Aside from that, once again decent running was crucial to eliminate one random battle (in the following segment though), which made this segment even more troublesome as I knew only after saving and doing the first 10 minutes of the following segment whether I was done with this segment.

Segment 12:
4 mandatory battles in just one segment, that's rather bad. Therefore, I most certainly didn't want to try risky experimental strategies here (but don't be afraid, you will see them soon enough). Still, several things had to go right. To begin with, Hundred Gunner or Heli Gunner (or Motor Ball and clearly Rufus) must not kill any character since reviving wastes time and items (and items are money), not to mention that I only had one Phoenix Down, and not reviving wastes turns (and hence time) and EXP. Moreover, Heli Gunner should not afflict anyone with Sleep (his AB Cannon has a chance of doing that), and neither should Dark Nation afflict me with its existence. So, Cloud's Braver has to hit Dark Nation before it is able to cast Barrier on Rufus, and Braver must also 1HKO Dark Nation, which proved to be problematic as Dark Nation has exactly 140 HP but, due to random variation of damage (1/16 of base damage), Braver did not all the time deal at least 140 HP of damage.
Finally, in the Motor Ball fight, Cloud's Bolt must deal enough damage so that, when all Grenades are used up, Barret or even worse Tifa don't have to employ their overly powerful Limit Breaks.

Segment 13:
Admittedly, this is one of the least exciting segments. At the same time, this segment is a rather aggravating one as it is both very long and requires quite some luck just at the very end of it. Basically, only a few things have to be accomplished here:
-The Dragon battle must be as short as possible. The Dragon has 3,500 HP and Sephiroth's physical attack is not streng enough to 1HKO it, so the Dragon will get a turn in between Sephiroth's. It goes without saying that, in case the Dragon kills Cloud, Sephiroth must not use Life 2. Interestingly, Sephiroth's physical attack is critical all the time. Besides, due to his AI, not only is he immune to both magical and physical damage, but also does Sephiroth not even use up MP for his spells.
-In the Mt. Nibel caves, running must be optimal so that no random encounters are triggered. Sephiroth is just too fond of attacking his opponents with those advanced magic spells that so incredibly powerfully waste my time.
-Finally and most importantly, after leaving Kalm, one must be able to run for a certain while and save before a random battle is triggered. I had to do this whole horrible flashback nine times because of this...

Segment 14:
This is fun. Well, it is if one makes it past the marshes without running into the Midgar Zolom (aka the Midgar Zolom skip). It took quite a while to succeed (around 10-20 resets per reaching the other side unharmed), however, as the segment is extremely short and running into the Midgar Zolom entails a fast Soft Reset, doing this was rather enjoyable.
Once on the other side, another skip should be performed, the Midgar Zolom Cutscene skip, which, when executed perfectly, that is opening the menu just once, saves about 12-15 seconds. The Midgar Zolom Cutscene skip is really not that hard and surprisingly consistent after a few minutes of practice. I made sure to run at the border of the mountain and always opened the menu when I reached a certain dark green line of the mountain texture.
Finally, after exiting the Mythril Mines, one still needs to run for quite a while in order to be able to reach Junon in the next segment without getting a random encounter. I was quite venturous in that aspect; I probably could have stopped running one second earlier, which would have spared me dozens of resets, but this would not have felt right without the appropriate amount of resets...

Segment 15:
Bottomswell is a very ulgy fish/bird/mutant as he, albeit being susceptible to poison, usually isn't poisoned after the first casting of Bio. Of course, that is wholly unacceptable. Furthermore, one should defnitely be able to throw a Grenade at Bottomswell before he gets his first turn. On top of that, Bottomswell must not use his (triple) Moonstrike attack, which is triggered if one of my characters' attacks reduces his HP to 1/4 or below of his maximum HP of 2,500 before a certain point of the battle. You can see that I was very close to that, which was the cause of quite an (un)healthy amount of resets...
Actually, I could have also used Limit Breaks (in particular Tifa's, which deals less damage than a Grenade) to make Moonstrike less likely, however, Limit Breaks cost around 5 seconds and I did not feel like losing time there since this battle is just at the beginning of this segment.
The death of Bottomswell is followed by the most exciting, adrenaline rush inciting minigame in the history of video games.
I probably would not save at upper Junon anymore today, but when I made this segment I was not aware of how easy the marching could be.

Segment 16:
This is a rather simple and easy segment but I still found ways to screw it up at the end...
The hardest part was defintiely running into that back alley before the marching without getting stuck on that red-clad soldier. In case you loathe the marching (as I used to), all you need to know is that mashing the [OK] button increases your score very rapidly. You can easily get up to 60% by doing this (Garland has a video on that) and I was actually going to stop at 1:33 as the last visible score showed only 49%.
Had I been able to position myself correctly onto the Save Point at the end, I probably wouldn't have lost 3 seconds.
I slightly regret not having redone this segment but I felt okay with it back then.

Segment 17:
I love this segment just as much as I hated making it.
In case anyone cares, this took me more resets than the elevator in the Shinra HQ.
Usually, if one is quite lucky, the Jenova-BIRTH battle takes around 2:30 min, however, I remembered how this battle worked out in a TAS and wondered if the same strategy could be applied in a segmented speedrun. Obviously, in order to do so, I would have to find a way to reliably trigger a Preemptive Attack in the Jenova-BIRTH battle, which I then did.
Here is what you need to know about Preemptive Attacks:
A (random) battle occurs as a Preemptive Attack if and only if the following prerequisites are met:
That battle has a Preemptive Attack flag (only some random battles and even fewer boss battles have these), the battle is initiated after a certain amount of steps (in a field with random encounters) have been completed.
Therefore, in order to get a Preeemptive Attack on Jenova-BIRTH, one needs to make sure that the fight occurs after a certain amount of steps, which is not as difficult as it may seem. All I had to do was to make sure that I reached Jenova-BIRTH while walking/running always the same amount of steps in the two screens inside the ship (only these have random encounters) and finding out which route to Jenova-BIRTH would yield a Preemptive Attack.
In order to accomplish the former, I never freestyled, i.e. I never changed directions while moving but only after coming to a halt. Pay attention to the part from 01:27 to 01:32 to see what I am trying to say. To accomplish the latter, I simply tested several routes, and the one you see in the video is the first I found.
Being able to trigger a Preemptive Attack on Jenova-BIRTH all the time, I "only" still had to reset a few more times (11 weeks worth of resets) to have Cloud get a critical Braver and Jenova-BIRTH use her most agreeable moveset...
Apart from that, the Cargo Ship is running hell: The ladder that leads to Tifa is just ridiculously brutal.

Segment 18:
Nothing all too cruel or unusual.
Just steal 2 Right Arms in 2 attempts...
Before that, I also skip a random battle by opening the menu at the same frame the battle would have occurred. I do that because it is very easy to execute this glitch here. As I simply hold down [Up] and [Cancel] for passing through the first screen, I can focus completely on watching for the right spot to open the menu at, and then mash [Menu] at 00:20.
Besides, I figured I could already practice this glitch since I intended to make use of it in a future segment...

Segment 19:
There is nothing one can do here to improve the chances of stealing Right Arms, however, one can still do quite some things here.
First of all, I did not grab the Turbo Ether yet because whenever I did, I ended up getting an additional encounter.
As I have already mentioned a few segments ago, random encounters on field maps are not random at all (neither "when" nor "which"). Therefore, I simply tested which random encounters I could get in the two screens I get random encounters in in this segment, and found that the sequence after 5 random battles after a Hard Reset yields the most agreeable outcome.
That's basically all. Oh yeah, and hundreds of resets...

Segment 20:
I buy a Pheonix Down for Gi Nattak because I need two due to the Repeating Boss Glitch.

Segment 21:
This segment is more annoying than it might seem.
First of all, there are basically two ways to reach Dyne.
One can follow the path to the uppermost screen and then turn right, walk through the gate and arrive at the screen prior to Dyne's. One can, however, also run left at said uppermost screen. Doing so will lead into a desert screen, where the next screen one will arrive at is randomly choosen. It is either the same desert screen or another desert screen with a huge chasm, which leads to Dyne as well.
I don't even know the odds of getting to that other desert screen on the first try but it took me about 15 resets per success.
The reason why one should even bother to choose that route to Dyne instead of the ordinary one is the fact that that desert path seems to be completely void of random encounters, whereas taking the usual way to Dyne almost certainly includes a random battle.
Then, there is Dyne, who, upon defeat, has a 50% chance of wasting about 10 seconds with his perfectly pointless final attack.
The Chocobo racing is quite annoying as well as it seems that the other chocobos' stats are subject to random variation. So, while winning the race itself is no problem at all, waiting for the 2nd Chocobo can sometimes take quite a while.
Finally, the distance between the Gold Saucer and Cosmo Canyon is so high that getting from the Gold Saucer to Cosmo Canyon without a random encounter and only saving once requires one to drive very far (and thus reset a lot of times...).

Segment 22:
I had a terrible time getting to Cosmo Canyon without a random encounter.
In fact, I had to do this ludicrously uneventful segment almost 100 times as I got stock on Cosmo Canyon's adorably shabby decoration the first two times I did not get a random battle.
Even though I could have been faster here, the 2-4 seconds I could have saved had my running been better just could not convince to do this segment again.

Segment 23:
This segment took me more than 4 times as many tries as I probably would have needed to do had I not been so cheap and deemed a certain recovery item dispensable...
Here is what you need to know about Gi Nattak:
Gi Nattak absorbs the Restorative Element, which is used for almost all means of in-battle recovery. Therefore, restorative items and magic damage Gi Nattak.
In fact, there are a lot of ways to 1HKO Gi Nattak using restorative items or magic:
Megalixir, Elixir, X-Potion, Phoenix Down (25% chance to hit), Life2, Life (25% chance to hit), FullCure and Angel Whisper.
Still, the way I beat Gi Nattak hardly stands out in this segment.
The fact that I beat him twice and skip a lot of battles should.
In case you are unfamiliar with the non-TAS Stinger skip, the random battle skip and the Repeating Boss Glitch, here is what you need to know:
1) The Stinger battles can be skipped by triggering a random battle on the same frame one enters the loading zone for the Stinger battle by touching (a certain part of) the spiderweb.
Obviously, triggering a random battle on a certain frame seems extremely hard outside of a TAS, however, it is not. Watch BrutalAl's video on that matter (http://www.youtube.com/watch?v=2BztApiaty0) or read on and I will explain it you.
2) Any random battle can be skipped by opening the menu on the same frame that random battle would have occured.
Once again, this skip seems rather difficult to execute but it is very easy in some cases. Also, I used that skip in Segment 18 already.
3) A mandatory battle can be repeated once if the previous battle was skipped.
All o this may appear to be relevenat only to a TAS.
It is not.
As I have said several times before, random battles in FF7 are in no way random if they occur on a field map (i.e. any map outside of the worldmap).
When a random battle will occur can be predicted most precisely.
Basically, a random battle is initiated if a so called danger value surpasses a certain threshould value when a so called battle check is performed (which happens around 4 times per second).
If the danger value is so high that a random battle is nigh, saving and soft resetting will reset it.
After saving and soft resetting (regardless of whether the danger value is reset or not), the danger value will always be the same when reloading the same save file.
As a result, one can run around in a field, note when a random battle occurs, reset and consequently predict that random battle.
Still, the problem is that, while this procedure enables one to foretell more or less accurately when a random battle will take place, one needs to know the exact frame the battle starts on in order to be able to skip it.
To cope with that problem, BrutalAl came up with an approach he calls "never to freestyle", by which he means not to change directions while moving but only after having come to a halt. If one employs this technique, not only will one know when a random battle will occur, but also will one SEE this, as random battles will, if one never freestyles, occur always at the same spot.
Granted, a lot of practice is required to be able to consistently press menu at any given frame, but hey, this is a speedrun, not a tea party.
Interestingly, I could have avoided the Repeating Boss Glitch had I forced another random encounter before Gi Nattak. Doing so and fleeing that battle seemed to be only 5 seconds faster, though. As I probably needed the EXP the two Stinger battles would have yielded, fighting Gi Nattak seemed to be a better idea (apart from the fact that this was just about the worst idea I could have come up with from a keep-your-sanity-inspite-of-ridiculous-amounts-of-resets perspective).
Besides, the Sneaky Steps must use Death Sentence on Tifa and leave behind (at least) 2 M-Tentacles.
I can't believe I still screwed up so horribly at the end and lost around 5 seconds...

Segment 24:
Nothing unexpected is to be seen here.

Segment 25:
Not much to say here, just the standard strategy.
It took me less than a day to make this, which is actually very fast.
Here's what you need to know about random variation of damage:
If a character or enemy uses an attack that allows for random variation of its damage (only a few attacks do not, e.g. Demi), then, at the end of the damage calculation, the base damage will be multiplied by a factor between 15/16 and 1.
What this means is that for every 16 HP of damage, there is one point of random variation.
For instance, if an attack deals less than 16 HP, it will (if it is not for a critical hit or several status effects) always do excatly that amount of damage.
If an attack does between 16 and 31 HP (before random variation is applied) of damage, say it does 25, it can deal two different amounts of damage (24 and 25 or 25 and 26 in the example).
If one hopes to inflict 99 HP of damage to Materia Keeper, 99 HP will be one of seven different outcomes. Also, if 99 HP is at the bottom or at the top of the damage spectrum, the probability to deal 99 HP of damage will be (slightly) lower than 1/7.

Segment 26:
This segment is made of text, running around, shopping and a boss battle in just about the worst possible order.
I considered going for Cait Sith' Dice in the Palmer battle. Had I even requested a 2,400 or 2,000 toss, this battle could have been over in about 25 seconds. But I actually did need Cloud's Cross Slash for Red Dragon, which means that I still had to use Braver two times against Palmer. Of course, this ruled out a (Greased Lightning-)fast Palmer battle.
Also, it meant that Palmer would have to attack Cloud exactly one time, and he did not really want to comply...
It goes without saying that the first or the second use of M-Tentacles had to posion Palmer as well, which did not work out all the time, either...
I ended up with a pretty decent Palmer battle. It's not exactly spectacular or exciting but I only spent two days on it, Cloud learned Cross Slash and Palmer died in around 50 seconds, which is all I cared for.

Segment 27:
The Gold Saucer Gauntlet...
Once again I need to say "This could have been an easy segment, had it not been for..."
And this time, the cause of misery is the Battle Square. Even though I flee the first battle, I could only accept fleeing from a Flapbeat battle, as I need to steal a T/S Bomb.
Of course, stealing a T/S Bomb is not that hard (I have a 40% chance) but there are actually a lot of different Flapbeat fights in the Battle Arena and, obviously, the more enemies, the more turns they get before I can steal from one and escape.
So, I decreed that only an encounter with a single Flapbeat and only if Flying Sickle is not employed shall be accepted.
Yes, I opted for a lot of pain only to gain 5 seconds at most...

Segment 28:
World map encounters are a very strange matter.
In a way they are more random than field map encounters, in another way they are even less random.
Basically, which field map encounters one gets depends on one's battle count since the last hard reset. Which world map encounters one gets seems to depend on one's movement on the world map.
As I needed to fight three Flapbeats to steal one T/S Bomb, fill Cloud's Limit gauge and win another T/S Bomb, I obviously wanted to get a Flapbeat encounter all the time, and I did.
If you watch closely, you can see that while running towards the Tiny Bronco, Cloud moves slightly to the right and to the left for a total of six times. Doing this allowed me to always get that Flapbeat encounter, even though I did not get it on the very same spot all the time.
Still, getting that encounter was not even half the thing...
Not only did I need to successfully steal a T/S Bomb in one try (40% chance) but also had the Flapbeats to attack Cloud often enough to fill his Limit bar (bad odds), and finally, they had to drop at least one T/S Bomb (1/3 chance).
I am thankful this segment is so short and I still had a Right Arm...
The rest is just getting the arguably most powerful materia in this run and skipping a random encounter (I can do this because I only run left in Gongaga until the skip).

Segment 29:
The Temple of the Ancients.
Behold the probably most major flaw in the run: I lose 18 seconds by talking to the black-caped Sephiroth clone because I simply did not know one could actually just run around him...
Apart from that, I merely had to learn how to navigate through the Escheresque first screen, manipulate a Side Attack encounter, master the minigame on the second screen, look up which path to choose on the screen before the Mural Room and engage in a jolly text skipping spree thorughout all that.

Segment 30:
Now that Tifa has Deathblow, the suffering may commence.
And by suffering, I mean the necessity not to miss a single Deathblow from now on until the end of the game...
What better prelude could there be than Red Dragon?
As I use Cross Slash to paralyze Red Dragon, which causes attacks targeting Red Dragon, inlcuding Death Blow, to gain an accuracy of 255% (the attack will (under almost all circumstances) not miss), I only have to hope Tifa's first Death Blow does not miss.
Red Dragon's first attack also has to be Red Dragon Breath and it must be used on Tifa.
All in all, it's a very manageable battle, unlike the conversations that precede it.

Segment 31:
I have made up a new strategy for Demons Gate. One that finishes Demons Gate before it ever gets to attack.
For this to work, Tifa must be (at least) on Level 19 and her Strength must be (at least) 43.
In addition thereto, one needs two Right Arms.
As Demons Gate has 10,000 HP, as a Right Arm does around 1,250 HP of damage and Tifa's Deathblow about 1,900 HP of damage under these circumstances, 4 Death Blows and 2 Right Arms will be just enough to kill Demons Gate.
While I was watching lolo262626' Speedrun Strategies video on Demons Gate, I realized that even when resorting to a so called "perfect tactic", one still needs at least 1:40 min to defeat Demons Gate.
I find this rather long. Even worse, a huge amount of that time is just wasted when Demons Gate uses Demon Rush and pulls itself back to the back of the screen after that attack.
Therefore, I looked for a way to defeat Demons Gate before it may use Demons Rush even once.
The obvious solution was to slow Demons Gate but, as I don't have access to Slow, I had to discard this idea again.
So, I studied Demons Gate AI and found something very intriguing:
Here's what you need to know about Demons Gate's AI:
At the beginning of the battle, Demons Gate will randomly set its turn count to an integer between 0 and 4.
If its turn count is 0 or 1, Demons Gate will start off with Falling Rocks; if it is 2, it will use Cave-in; if it is 3, it will use Petrif-Eye or Falling Rocks.
Moreover, its turn count will be incremenetd by 1 after any of these attacks and randomly reset to 0, 1 or 2 once it reaches 3.
More interesting is what happens if its turn count is 4.
Namely, it will use an attack called Advance, which is actually just an attack animation (it will swing around its arms and move closer to the characters).
Now comes the intriguing part.
If Demons Gate's turn count is 4, there is a 1/3 chance it will be set to 5 and a 2/3 chance that it will be set to 6.
If Demons Gate's turn count is 6, Demons Gate will use Demon Rush and its turn count will be reset to 0.
If it is 5, on the other hand, Demons Gate will do nothing and just increase its turn count to 6.
You can probably already guess what Demons Gate's optimal behavior would be.
But there's even more.
The first time one reduces Demons Gate's HP to 7,500 or below, Demons Gate's turn count will also be set to 4.
If one employs the Wait Trick and uses the information on Demons Gate's AI, it is very well possible (provided all Deathblows connect) to beat Demons Gate in only 50 seconds.
Here is how that works:
Demons Gate's inital turn count is 4 (which happens with a probability of 1/5), and it is then set to 5 (1/3). Meanwhile, Aerith casts Death Sentence on Tifa and Tifa hits Demons Gate with a first Deathblow. Demons Gate gets another turn but does nothing, its turn count is set to 6 though. Tifa hits Demons Gate with another Deathblow. As Demons Gate's HP is thereby reduced to less than 7,500, its turn count is set to 4.
Demons Gate gets another turn but only swings its arms, its turn count is once again set to 5 (1/3). Cloud throws a Right Arm and Tifa damages Demons Gate with a third Deathblow. Demons Gate gets another turn but does nothing. Its turn count is set to 6, Demon Rush is imminent. Aerith throws a Right Arm and Tifa successfully uses Deathblow on Demons Gate for a fourth time. Demons Gate dies.
So, the probability of the optimal battle is 1/45 on Demons Gate's side. Unfortunately, it seems to be much lower on Tifa's side (all 4 consecutive Deathblows must hit; if you enjoy resets, this will be fun).
Combining these two requirements, one likely ends up with odds of around 1/1,200.
Thankfully, it takes less than 30 seconds to fight Demons Gate again after a soft reset.

Segment32:
It could have been a nice little glitch fest. Had it not been for the outrageous amount of running mistakes. This time, they add up to nearly 5 seconds...
On the other hand, I was able to pull off the chest opening animation skip, which is triggered by mashing the [OK] button before the screen fades in again.
It's not a huge skip but it saves around 1.5 seconds.
More interestingly, I do the paralysis dodge skip in the City of the Ancients. This skip saves around 15 seconds and is triggered by opening the chest in the house with the beds from the 2nd floor, which is a glitch in itself, at the same time the "Do you want to rest?" dialogue comes up.
Finally, the Jenova-LIFE battle went smoothly. She had to open the battle by attacking Tifa and Cloud, so that Tifa's HP would be in critical range and Cloud would get a Limit Break. As soon as Tifa did not miss the first Deathblow, the battle was done (the following two were sure to hit as Jenova-LIFE was then paralyzed).
By the way, I took Red XIII into my party instead of Cait Sith because Red's reaction to Aerith' death is more than 13 seconds shorter than Cait's.
Incidentally, Yuffie has by far the longest of all.

Segment 33:
Basically, all one has to do in this segment is to run almost all the way to Icicle Inn without getting a random encounter, which is much easier said than done.
The distance between the exit of the Coral Valley Cave and Icicle Inn seems to be almost as high as the distance between the Gold Saucer and Cosmo Canyon.
Fortunately, this segment lasts for only 2 minutes, so it didn't really matter that I had to replay it more than 50 times before it worked out the way it should.

Segment 34:
This segment was blatant horror.
Usually, there is any one thing that can go wrong in a segment, for example failing to steal a certain item in one try, being unable to skip a random battle, screwing up a minigame, getting into (too many) world map encounters, wasting too much time menuing or getting stuck while walking around.
But not so in this segment. In this segment just about everything can go wrong.
First of all, one has to make it to Icicle Inn without a random battle. As simple as that may sound, it was horribly aggravating due to how world map encounters work. They are triggered depending on one's movement on the world map.
While this may suggest that one can exploit this by finding an encounter-free route and then simply taking this route in all following attempts, which seems to be indeed possible, one has to replicate that route very accurately for this to work, which proved to be almost impossible here.
I did not keep track of it, but it feels like I had to spend on average at least 10 attempts in order only to arrive at Icicle Inn in the first place.
I grabbed the Vaccine because I will use it in a wicked boss battle in a later segment. Besides, it takes only one second to pick it up.
The next obstacle in this segment was the snowboarding.
I got quite consistent at it after practicing it for twelve hours without interruption. Still, I only continued here when I made it to the end sub 2'30 and crashless, which was the case half of the time.
After the snowboarding, more fun in the snow shall commence.
Like the Cave of the Gi, Great Glacier can be completed without freestyling. In fact, it is even easier to find a suitable route.
Still, the last battle skip on that nearly pitch white snow field screen was rather difficult to execute.
The source of greatest pain was without a doubt the Snow battle. Stealing that Circlet in one attempt is a rare occurrence, even though it is much more likely (around 8%) than stealing two Right Arms in two attempts was (around 1%).
However, the two Right Arm stealing segments were composed almost exclusively of stealing these two Right Arms. This segment on the other hand is not at all. Getting to this very battle in one out of dozens attempts only to reset here was no fun at all.
But that's not even all.
As I said, there is still a rather difficult random battle skip after Snow and the entire snowfield passage.
Now, that snowfield is regarded as part of the world map, which means that encounters occur depending on one's movement, which is very bad. Sometimes I passed that area without any random encounter at all, sometimes I got three.
Initially, I was going to accept only an attempt without a random battle in the snowfield but that seemed just too frustrating, and I decided to accept up to one random encounter if I would escape from it very quickly.
Finally, there is Mr. Holzoff's hut, home to one of the most annoying savepoints so far and to a short menu sequence. Unfortunately, my menuing was far from perfect but I was not in the right mood to spend another month on a 3-second improvement at best.
Sorry for the framerate overlay that pops up a few times. I turned it on to see how much the snowboard minigame was lagging (I had to reset twice because it slowed down so much I crashed into a wall) but forgot to turn it off again.

Segment 35:
This segment may look exciting.
Apart from learning Magic Breath in a Stilva encounter, though, this segment mainly consists of traversing the first half of Gaea's Cliff, which is not exactly that exciting.
In fact, of the three outside screens in which Cloud's body temperature has to be kept above 26°C, only the second one necessitates any warming up (for about 2-3 °C) at all.
I don't know if even more walking could have reduced the number of random battles even more but the route I chose allowed me to skip two random battles, which already seems like a lot.
Actually, I was considering to menu-skip all random battles except for the Stilva one, however, even though this was possible, I took longer than just fleeing from the battles and it was almost impossible as I ended up having to skip 12(!) battles.
Now, the Stilva fight worked out perfectly as Stilva started off with casting Magic Breath (which happens with a probability of 25%).
I read that due to Cloud's stats, one might have a hard time surviving Magic Breath and 1HKOing Stilva afterwards but, fortunately, I survived Stiva's Magic Breath and was able to kill Stilva in one hit afterwards all the time, which made this segment much less annoying than it could have been.

Segment 36:
I was going to end the speedrun here.
Not so much for the Icicle battles. Although they are very luck reliant, there's a simple way to make matters much less homicidal.
The reason why this segment was so much more frustrating than anything before it is the fact that I had to redo it.
Well, so far, I had to redo every segment. I don't recall having finished a segment in just one try yet. This time, however, I had to redo an almost impeccable segment.
But not just once. 14 times. And not for a stupid mistake. But for Tifa's stupid stats.
While I was practicing the Schizo battle, I realized that for a certain strategy to work, Tifa's Strength and Spirit would have to be in a certain range. You can probably already guess the gist of the story.
Yes, I had to redo this highly annoying segment, save, check Tifa's stats (her Spirit has to be high enough to enable her to survive some of Schizo's attacks and, even worse, her Strength must be neither too high nor too low, or else my strategy won't work), reset, repeat this for 9 times, doubt my ability to ever achieve this, take a long break, die a little bit on the inside, repeat it for two more times, consider changing my Schizo strategy, consider how to explain I would discontinue the run, take another break, try it two more times again, become aware that I had eventually made it.
On to the Icicle battles.
I have honestly no idea how anyone of sane mind could attempt the three mandatory Icicle battles with active ATB. Basically, ATB on active pretty much guarantees that all Evilheads will get at least(!) one turn. If they use Ultrasound on Cloud, the segment is over. If they kill anyone, it is as well.
Therefore, one should do as much as possible to reduce the number of turns they get. ATB on wait does exactly that.
As I had a spare T/S Bomb, I found a very nice way to make use of it and save quite some time.
Since the Icicles are weak to Gravity; Demi2, Demi3, Laser or a T/S Bomb will instantly kill them, unless of course, one uses any of these attacks, except for Demi3, after having used another attack on the Icicle and having reduced its HP to an odd number beforehand. In that case, it will be left with one 1 HP after Demi2, Laser or a T/S Bomb...
Yeah, that's so cool...
Apart from that, Demi only has a MAt% of 75%, which was the source of hundreds of resets. Moreover, the Icicle has a 50% chance to counter any non-fatal attack by using Icicle Drop on a random party member, which was the source of even more resets.
I accepted up to two Icicle drops in total, if they did not kill anyone. Unfortunately, they were lethal around half of the time.
I was getting two Icicle Drops most of the time, followed by one and three Icicle Drops (three were a certain reset because of deaths, though). Never did I get 4 or 5 Icicle Drops and only a few select times was I able to go without any at all.
I was incredibly relieved after I had realized that, when I was witnessing an Icicle Drop-free session again, I would finally not have to redo it ever again.

Segment 37:
This was probably the easiest segment so far.
One merely has to get up to Schizo's lair, escape one random battle, which turned out to be a Preemptive Attack for me, thus making the escaping part as easy as it can be, and finish with some light menuing exercises.
I split this segment from the previous one for three reasons:
Firstly, saving after the Icicle session removes a random battle on the way up to Schizo.
Secondly, the Icicle segment was purely evil, so I would not hazard trashing it for something as common as a menu mishap or getting stuck anywhere on the climbing screen.
Finally, this segment is right in the middle of two evils, and having this segment as buffer in between allowed me to try out different strategies for Schizo and adjust my equipment accordingly without having to do this at the beginning of the Schizo segment.

Segment 38:
SCHIZO.
There really shouldn't be any more to say than that.
Let me begin with some important details on Schizo.
Either side of Schizo, they are actually called "Schizo(Right)" and "Schizo(Left)", has 18,000 HP and nullifies Gravity damage. Schizo(Right) absorbs Fire attacks, while Schizo(Left) absorbs Ice attacks.
Both sides work on a rather simple AI:
They start off with using their respective signature Breath moves, which are Fire- and Ice-elemental respectively and which they will do for the first three turns.
The following two turns, they do nothing at all.
Ideally, they have to be dead by then because their next turn will be spent on a highly damaging attack spree before they resume their ordinary Breath moves afterwards.
When either side dies, it uses a Lightning-elemental final attack.
Additionally, either side will counter-attack with an Earth-elemental ability upon having been targeted for the fifth time by a character's attack.
Obviously, one cannot tolerate that Earth-elemental counterattack. Hence, Schizo(Left) and Schizo(Right) each have to die by 5 attacks at most.
As one can tolerate their 6th turn's attack even less, they also each have die in 5 turns at most.
What this means is that one will see exactly 8 Schizo moves in this battle, 3 Fire Breaths, 3 Ice Breaths and 2 Lightning final attacks.
Cloud's Dragon Armlet halves Fire, Ice and Lightning damage.
Red nullifies Fire damage and absorbs Lightning damage due to the Fire Ring and the Bolt Armlet.
Tifa halves Lightning damage because of Ramuh=Elemental in her armor.
As a result, in order for my strategy to work, Schizo has to act in a certain way:
Exactly one side must hit Tifa in the first turn. She must not be targeted again by the next four attacks.
No one is allowed to die throughout the battle. Cloud can sustain up to three attacks of either side, while Red can sustain 1 Ice Breath and any amount of Fire Breaths.
More interestingly, Tifa and Cloud can each survive two final attacks, Red is healed by them.
Apart from the restrictions on what Schizo may do, Tifa has her very own list of requirements.
To begin with, no Deathblow may miss. Less obviously, none of her Added Cuts may become critical. And just as horrible as that already is, for good measure, the random variation of Tifa's attacks must be favorable.
She might do so much damage that two Deathblows and Added Cuts kill either side of Schizo, which nets a reset. On the other hand, she might not do enough damage, allowing either side to suvive Trine...
Very rarely, that only happened once, Tifa might die by the final of the two final attacks and miss out a lot of necessary EXP.
This is a horridly atrocious battle.
At least, I was fully aware of that from the beginning, so I knew what I had to expect...

Segment 39:
The Jenova-DEATH segment could have been an easy one.
Yeah, could have been...
Jenova-DEATH isn't a particularly demanding foe. To be precise, only 4 not so unlikely things most occur:
Jenova-DEATH must open the battle with Silence, which must not hit Red XIII (it must hit/miss someone else or miss Red XIII), and Tifa's first two Deathblows must connect.
In fact, had I not forgotten to equip Red with the Ribbon in the last segment or deemed the additional time it takes to equip him with the Ribbon now (around 3 seconds) an utter waste of time, the battle would have been even less luck relying.
Also, I could have used the Hero Drink I saved up in the Schizo battle here, which would have allowed me to kill Jenova with only two Deathblows. However, that would have saved less than 10 seconds and even more time can be saved if I use the Hero Drink in a later fight.
The reason why this segment still was not as easy as the Jenova-DEATH battle would suggest (actually, the Jenova-DEATH battle was not THAT easy to begin with but compared to Schizo, it most certainly was) is that I only accepted attempts in which I was able to flee from the two random battles right away without being attacked.
Not only did getting attacked force me to reset, whenver Cloud or Tifa were killed, most of the time anyway but also did Farringa pretty much already have a perfect segment.
Therefore, in order for me to cut any time at all, I could only do this by getting extremely short random encounters. I tried skipping them as well but I ended up spending too much time on getting through that screen in a non-freestyle way.

Segment 40:
This segment is even more sleep-inducing than the Kalm part. It is slightly shorter than the Kalm flashback, however, whereas the Kalm flashback did at least contain one random battle (with Mr. Hojo junior), this segment does not. In fact, this segment consists of very little running, some FMVs and primarily piles of text skipping.
All one needs to do or make sure here is:
-Don't get into a forced encounter in the last Whirlwind screen. Obvious, right?
-You can save an incredibly huge amount of time, 0.2-0.6 seconds to be precise, if you move Cloud to a certain spot in the screen in which he is walking upside down on the ceiling so that he won't have to walk closer to the crystal, holding the upper half of Sephiroth, who has mysteriously begun to grow roots as replacement for his lower body, when the cutscene of Cloud handing the Black Materia to Sephiroth is initiated.
-You can lose a considerable amount of time, 12-18 seconds, in the first screen in Junon with Barret and Tifa. After Barret has shown Tifa Meteor by the window, Tifa is supposed to talk to Barret to make Rufus enter the room. However, if Tifa is too close to Barret before the looking through the window has taken place, it will be impossible to talk to Barret afterwards for a seemingly random amount of time. I don't really know why this happens but it's a fairly well-known issue.
-Be good at text skipping.
Now, text skipping is just mashing [OK]. I might be a tad too obsessive about text skipping since I have developed a technique that involves using both index and middle fingers. It does work quite well, even though it looks just ridiculous and has increased the wear of my keyboard by around 300%. Yet, there is an obvious downside to this technique as I can't perform high-tier text skipping for more than about 12-15 minutes without breaking my keyboard or my right arm.

Segment 41:
This is not a very good segment:
First of all, how horrible is this segment?
According to my estimations, I lose between 7 and 9 seconds in this segment due to the mistakes I made.
Strangely enough, this segment is supposed to be a very easy one. There's not really much randomness involved, it includes merely 3 encounters, 2 random and 1 fixed, all of which can be escaped from.
Aside from that, it's mainly 10 minutes of text skipping and a lot of FMVs of Junon and Sapphire Weapon, and Tifa's gas chamber adventures, which I am amazingly consistent at screwing up. I don't even know whether it's a problem pertaining to the PC version of this minigame or my grossly inexistent abilities at this quest (which I deem the more likely explanation).
Anyway, I could never get her to use the key she's holding with her teeth, immediately on her arms without making a useless movement with her head first.
I lose 2-3 seconds because of that.
The other main source of time loss, in addition to the one resulting from my inability to land the Highwind close to Mideel quickly is running mistakes.
I had hoped that running would never again be so evil after Midgar but I guess I was wrong. Running around on the Highwind is almost worse and allowed me to lose another 4-5 seconds...
Still, I decided to keep this segment because I got very lucky with the encounters. I could flee from the first one right away and from the following two after one attack.
As I had a lot trouble with the first encounter in particular, escaping from it before being attacked even once, which had never happened in my previous attempts, saved 6 seconds at least.

Segment 42:
This is a very agreeable segment.
In fact, it is one of these very rare segments that does not include any kind of randomness. On top of that, it's not corrupted by an endless talking, text mashing, blah blah blah spree (I really hate dialogue). Instead, there is a swift shopping spree and only some light text skipping.

Segment 43:
This was intense and annoying.
Once in a while, hitting you like a cold, hard, used snowboard, a segment is hard not because of its randomness but exclusively because of the technical perfection required.
This is exactly what can be said about this segment.
As a a matter of fact, there is no randomness at all involved. As long as I am able to skip the first 2 and the last 2 battles, everything is fine.
I manipulated the third battle in between those skips to always be a Side Attack as skipping that battle as well would not have been efficient (I ended up having to skip another battle just seconds afterwards).
Even though skipping random battles is generally not THAT hard when abstaining from freestyling, it most certainly was here, and that first wooden bridge screen is why.
Normally, when I make up a route for skipping random battles, I start by finding a non-freestyle route before consequently optimizing and memorizing it. Afterwards, I take that route for the first time and run until the first random battle occurs. I escape from it and make a screenshot of the exact position Cloud/Tifa/Cid is at after the battle in order to open the menu on that same position the next time. From then on, I simply iterate that procedure until I have tracked down all random battles on that route.
Finally, when I know of all the spots to skip random battles on, I start practicing the skips until I can consistently execute them at least half of the time.
Unfortunately, this was much easier said than done this time because the wooden bridge screen is so terribly zoomed out that merely finding the right spots turned out to be basically looking for pixels. Even worse, knowing the right spots is only the easy part, for being able to open the menu on these spots with such a horrendous camera position was more or less just based on guessing. Even after hours of practicing the wooden bridge screen, I could skip the first two random battles in only one out of five times or so.
Unfortunately, even still worse, in order to force a side attack in the third battle, I had to make sure the battle count since the last hard reset was equal to a certain number. What this means is that I could not simply get another attempt in just 20 seconds after a failed one. Instead, after every failed attempt, I first had to do a hard reset, load my file, get into 5 random encounters and trigger a soft reset by dying in the fifth random battle before I could try this segment again.
Yeah, it took me twice as long to set up the segment as the actual segment lasts...
Additonally, two more random battles had to be skipped and a rather difficult menuing sequence was yet to be performed at the very end.

Segment 44:
First of all, setting ATB to Wait makes things much more feasible.
I could have done that in the previous segment but as I had to skip a random battle in this segment anyway, I did not lose time by adjusting the ATB just in this segment.
Basically, there are 6 mandatory battles that must be kept as short as possible. I was trying out different strategies and, to my great horror, I realized that I would have to make use of Cait Sith's Dice to save quite some time.
Unfortunately, as I had used one T/S Bomb in the Icicle battles, I was left with only two more. Although two are enough to get through the Train Gauntlet, two aren't quite enough to get through it quickly. Therefore, I had to find a means of inflicting around 2,000 HP of damage to Wolfmeister without wasting a lot of time.
As it turns out, Cait Sith is capable of doing precisely that, using his Dice Limit Break and getting a double 5 or a double 6 toss.
While the odds of 1/18 of that event might not seem too bad, this segment was overall a serious hardcore pain.
Obviously, not only does Cait Sith need to get the right outcome in his Limit Break but also do a lot of other things need to work out right.
To begin with, catching up with the other train as quickly as possible is almost as bad as getting a very good time on the Snowboarding minigame. In fact, I had to spend 2 hours just practicing the catching up part. On the other hand, practice really payed off there.
Additionally, in order to finish this section as fast as possible, one needs to be pretty lucky with one's characters' and the enemies' ATBs.
The Gas Ducters' Smog Alert and Eagle Gun's general behavior were the main source of resets next to those due to Cait Sith's generic incompetence.
What's more, Eagle Gun might even kill a character rather easily.
Finally, this segment could have been nice had it not been for my failure to stop the train without losing around 2 seconds because I forgot what buttons I had to press at the end...

Segment 45:
This segment has it all.
A Weapon battle, a minigame that's easy to exploit, a lot of menuing and, most importantly, Tifa talking mindlessly seemingly endlessly for approximately 20 minutes.
Now, this is certainly not an incredibly hard segment, since there are only three things that must go right, but due to its sheer length, it was bound to not be much fun.
As I said, merely three conditions must be met in this segment:
(1) I have to be able to survive the Ultima Weapon battle. In case you didn't know, Ultima Weapon will flee after three attacks, which will always be Quake2, his physical attack and his Unknown-elemental (yes, that's an actual element in FF7, it's also known as Hidden-elemental) Ultima Beam in that order, so I don't even have to attack Ultima Weapon.
(2) I have to be able to steal the Curse Ring on the first try. While I could have been able to go for a second or a third attempt, failing to steal on the first try wastes at least 3 seconds. As the Fort Condor section is as simple as it could be and since the Ultima Weapon encounter takes place so early on this segment, resetting until I was able to steal the Curse Ring in one attempt was not that nightmarish (even though getting (2) right did not necessarily mean getting (1) right...).
(3) I have to be able to not screw up. This was not as easy as it may sound. The menuing, the running around and the text skipping can certainly wear one out.

Segment 46:
This is a comparatively easy segment.
There are only two forced battles in this segment and, thanks to ATB on Wait, it is not too hard to finish them quickly. If Tifa does not miss either of her two Deathblows and the Attack Squads do not get to attack more than once (it is even possible, albeit quite rare, to end both battles without ever getting attacked once), the segment is more or less golden.
What follows afterwards is just a threefold random battle skip, one random battle and some light menuing.
By the way, I do not skip the fourth random battle because doing so would not have saved time. Skipping random battles does not reset the so-called danger value, and whether a random battle occurs on a field depends on that danger value.
Apparently, after I skipped the third random battle, the danger value was so high that a random battle would occur almost instantaneously. I actually tried skipping the fourth random battle but I ended up getting another random battle less than a second later. Therefore, getting into the third random battle instead of skipping it actually saved time.
Even though escaping from random battles is allegedly harder with ATB on Wait than with ATB on Active (I am not even sure if that's true), changing the ATB settings for that one random battle did not help save time in my practice runs.

Segment 47:
Too many battles. For just one segment.
As a matter of fact, there are 6 battles in this segment (2 before Carry Armor, Carry Armor and three afterwards) and all of them have to turn out almost perfectly.
To begin with, right in the beginning I had to correct a small mistake. As I realized that Tifa having a limit would cost me some time (since she can then no longer use her attack command), I had to reset her Limit bar just now. It would have been better had I done this at the end of the last segment but I was not in the mood to redo the previous segment for a 2 second gain.
Also, I can't believe I am saying this but it seems I am much worse at the submarine minigame than at snowboarding. In fact, my submarine time in this segment can only be called horrible as I lose 4-5 seconds compared to Farringa's run. On the other hand, unlike Farringa, I do not save before the submarine section, which allows me to regain around 3-4 seconds.
Still, as the 6 battles did indeed turn out almost perfectly, I did not even consider redoing the segment for the sake of a better submarine time and a merely very slightly better overall time.
Now, what can go wrong in those 6 battles?
About anything.
How likely is that?
It's almost certain.
What can one do to still get a good time?
Repeat it.
For a few hundred times.
First of all, Carry Armor isn't even the biggest problem in this segment. As long as he does not start off with Lapis Laser (50% chance), he is quite safely doomed, provided Tifa complies and does not miss a Deathblow, which is sadly not that likely.
The first two battles aren't much of a problem because they are so close to the savepoint. Therefore, accepting only attempts in which the Submarine Crew does not kill anyone and the Underwater MP don't get a turn before Tifa, who shall not miss her Deathblow, kills them, was not too annoying.
(Once again, ATB on Wait helps reduce the number of resets by a factor of 5-10. I don't even know why I ever tried doing forced battles on Active to be honest.)
Almost the opposite must be said about the remaining three battles. As Cloud's and Cid's MP are scarce, I cannot use Magic Breath in all three battles. Obviously, I didn't want to risk missing a Deathblow so late in the segment either. As a result, I had to settle for Trine, which caused a lot of pain because Trine may not be able to kill a Submarine Crew (having 1,500 HP) by only a few HP due to random variation of damage. Even worse, the final Pincer Attack usually entailed a lot of deaths, of my party members that is...

Segment 48:
This was nearly as bad as the previous segment.
Frankly, it was worse.
Whereas the previous segment had 6 battles, much more text to skip and even a minigame, this one only has 5 battles, which are even all at the beginning of the segment, and is basically done after the last battle.
Still, there are even more things that must be accomplished throughout these 5 battles.
Firstly, this is the last serious chance, albeit not the last chance, to get Limit Breaks, and I need to have two for Diamond Weapon.
As I reset Tifa's Limit bar in the last segment and since she is in critical health almost permanently, she probably won't be able to get a Limit Break.
Thus, Cloud must be attacked often enough for his Limit Break gauge to fill up. Without dying by the way...
Secondly, I need to steal the Ziedrich. In as little attempts as possible. In one attempt.
This is even less likely than stealing Ultima Weapon's Curse Ring in one attempt.
Finally, in order to save as much time as possible, the enemies must get as few turns as possible. In particular, one can and should finish the first two battles without being attacked at all.
In case you forgot, Tifa's Deathblow is (still) likely to miss, and a Deathblow miss is an immediate reset on my list, as is a character's death, an enemy's choosing an attack with a fairly long animation (like Grenade) or just a generic non-trivial screw-up on my side.
I lose a few seconds at the end of the segment because I could not remember where to locate Cosmo Canyon on the world map. Great, isn't it?

Segment 49:
This is an easy segment.
It's also the last easy segment...
Basically, it consists exclusively of text skipping and running around.
There is not much else to say.

Segment 50:
This is where everything falls apart. In a way.
Normally, one is merely supposed to wait for Diamond Weapon to reach the shores of Midgar, which is where one can then engage in the Diamond Weapon boss battle.
It takes Diamond around 135 in-game seconds to arrive there.
He advances as long as one is oneself on the world map. If one is not on the world map, Diamond does not move on.
Therefore, I obviously lose some time in this segment because I spend time on field maps. However, the time I lose (around 40 seconds) is spent extremely well as I will be able to make up around 4 minutes with the 40 seconds I "waste" here.
As you probably already know, I will skip the return to Midgar, which, albeit being utterly time-saving (about 12 minutes are saved), means that I will not acquire Cloud's Ragnarok. This can be a very detrimental consequence of the Midgar Skip since the Ragnarok is Cloud's second best weapon, not only in terms of Attack but also due to its magic boost.
To be precise, the Ragnarok increases Cloud's magic stat by 43, which is surpassed only by the Ultima Weapon (+51) and matched by the Apocalypse.
Obviously, I cannot get the Ultima Weapon or the Apocalypse but Cloud still needs (it later turned out he actually does not) the magic boost. So, I decided to get the Heaven's Cloud, the magic boost of +31 of which makes it Cloud's third best weapon for that purpose.
Also, I get the Heaven's Cloud because I had to visit the Gelnika anyway.
While Tifa is no doubt the most potent damage dealer in this speedrun, one can improve on that.
What's better than having Tifa in your party?
Having 1.5 Tifas in your party.
Cid, equipped with his Venus Gospel and the Ziedrich (+20 Strength), will in fact be able to deal around half the damage Tifa is capable of inflicting.
But there is even more.
Added Cut=Deathblow is nice, but 2xCut is almost as good, and it's just within reach.
Fortunately, I was able to enter and exit the Gelnika all the time without getting a random battle (escaping such a battle would have been almost impossible).
Had I got a random battle there on the other hand, I would have had to save inside the Gelnika first and split this segment into two in order to make that random battle disappear, which would have cost me around 3-4 seconds.
Finally, there is menuing at the end, which I have grown quite fond of.
In between, there is still a tiny detail. Getting two War Gongs in exactly one world map battle. It was worse than stealing two Right Arms in exactly two attempts 30 segments ago.
As I had to get into the Grand Horn battle quickly, I had to run around like a maniac, which means that I could not even make sure to get that 2 Grand Horn battle all the time, and I ended up getting that 2 Grand Horn encounter in only 1 out of 4 battles on average.
Getting two War Gongs is even worse, the probability of winning two is around 2%.
So, on average I should have had to spend 200 attempts to win two War Gongs in one 2 Grand Horn battle.
I had to spend definitely more than twice that number of attempts before it worked out the way it should...

Segment 51:
Diamond Weapon.
In order to save time in this fight, several things must go right.
First of all, Cid or Cloud must be able to use a Limit Break before Diamond gets his first turn. This is much MUCH harder than it may seem as Diamond Weapon has a Dexterity stat of 180. Afterwards, Cloud or Cid will be able to use a Limit Break as well due to the precedence of Limit Breaks over usual attacks.
Hence, Diamond Weapon will then use Countdown and become susceptible to physical attacks. Provided everything that follows turns out perfectly, Diamond Weapon will be able to attack a character twice in the course of the battle. He may not attack Tifa even once, Cloud only once and Cid up to two times.
After Diamond Weapon has started his countdown, Cid and Tifa have to attack him in that order.
As Diamond is weak to Lightning-elemental damage, having Ramuh=Elemental in his weapon allows Cid to do double damage to Diamond. Cid's two Deathblows must hit, which is not that unlikely thanks to the nice Def% boost of the Ziedrich.
However, the main problem is Tifa's 2xCut. Diamaond Weapon will only die before his third turn, which is more than desirable, if one of Tifa's attacks becomes critical. The Curse Ring gives a +10 boost to Luck, which is just enough for a critical hit to become possible in the first place. The probability of at least one critical hit by Tifa is as high as 6%.
As soon as Diamond dies, it's FMV time until the end of this segment.
The menuing at the end is preparation for Hojo in the next segment.

Segment 52:
The Midgar Skip and Hojo.
If you want to try out the Midgar Skip yourself, which is incidentally the (currently known...) second most major (after Yuffie Warping) and the easiest skip in the game, just hold [DOWN], [RIGHT] and [RUN] down when the scene fades to Hojo starting at 00:50.
The most important thing in the Hojo battle (and in all the remaining boss battles) is ATB on Wait. It will (among other very beneficial effects) allow one to skip Hojo's capsule attack Much more often (at all).
I am very glad I spent 1 second on grabbing the Vaccine in Icicle Inn. It's extremely useful in this battle, as is the War Gong.
Berserked characters will do 1.5 times as much damage. Additionally, their chances at getting a critical hit are entirely unchanged. Also, they choose their enemy targets randomly, which is slightly bothersome for the second part of the battle against Hellectic Hojo and his two arms since Tifa seems to take greatest pleasure in attacking Hojo's arms instead of his main body.
In order for my strategy to work, Tifa has to hit Hellectic Hojo critically at least once, which happens with a probability of around 5%.
Hellectic Hojo may choose any of his attacks before his early demise as long as it does not hit Tifa or Cloud.
Finally, Lifeform-Hojo-N's Combo attack must not hit Tifa or Cloud and he must not afflict Tifa with Slow.
Lifeform-Hojo-N dies after its first attack because I make use of the Wait Trick. Cid is only attacking Lifeform-Hojo-N for that very reason, it didn't even matter if his Deathblow would hit because Tifa's twofold 2xCut deals enough damage (32K+) to kill Lifeform-Hojo-N (30K HP) regardless.

Segment 53:
Let's have a glitch.
Better yet, let's have two of them.
Here's my checklist for this segment:
-Trigger the opening of the Save Crystal chest on the same frame the landslide animation occurs to save 1-2 seconds.
-Trigger the Save Crystal split-up glitch by trying to use the Save Crystal while descending the second stalactite to save more than 30 seconds.
-Send Cait Sith, Barret and Red XII to the right. Cait Sith is to take the right path in order to grab an Elixir.
-Send Tifa left and then up to grab a Hero Drink.
-Make sure Tifa dies before she leaves the party.
-Make sure the enemies get as few turns as possible in the random battles.
-Make sure neither Cloud nor Cid dies.
-Use the Save Crystal, save, rejoice.

Segment 54:
This is the final segment.
It is made up of the Final Descent, the three Final Boss Battles, Jenova-SYNTHESIS, Bizarro-Sephiroth and Safer-Sephiroth, and the Final Battle, Sephiroth.
Firstly, I need to grab Tifa's Hero Drink and Cait Sith's Elixir before I leave for the Final Descent.
The Final Descent consists of two screens with pseudo-random battles. Whenever Cloud jumps to another plattform, there is a 76/256 chance (roughly 29.7%) a battle will occur.
If a battle ocurs, it will be either against an Allemagne (26/256 chance), a Dragon Zombie (102/256) or an Iron Man (128/256).
13 jumps have to be performed to reach Jenova-SYNTHESIS, and it is perfectly possible to accomplish this without triggering any battle at all. The probability of that event is (180/256)^13, which is equal to slightly more than 1%.
By all means, do not accept more than 1 battle on the Final Descent (6.6%). Better yet, do not accept any battles at all.
Just getting into those battles wastes time but, even worse, you will probably end up severely damaged, in particular if you encounter Iron Men, which makes even more menuing necessary.
On top of that, it is almost impossible to escape from these battles quickly. If you encounter Dragon Zombies, you will most likely have to witness two attacks and lose at least half a minute as Dragon Zombie only has attacks with aggravatingly long animations.
Also, this is the final segment. You have all the time in the world to get this right. So drink your goddamn TEA and take your time.
Actually, how many battles one gets on the entire Final Descent is determined by the time one leaves for the first Final Descent screen. I did not take the time to analyze this most thoroughly but simply spent a few days on spending slightly different amounts of time before leaving for that screen...
Jenova-SYNTHESIS is the first final boss and already the most difficult with my strategy. As soon as her HP drops from 60,000 to below 15,000, she will start her Ultima countdown and not attack again before the counter reaches zero.
The fastest way to inflict more than 45,000 HP of damage to Jenova is by having Cid Deathblow=Added Cut her for more than 9K HP and having Tifa 2xCut her twice under the Berserk status for more than 19K HP each.
When a character is afflicted with the Berserk status, they will choose their target randomly. There are three enemies in this battle and Tifa's first two 2xCuts must hit Jenova-SYNTHESIS A, which happens with a probability of 1/9. Cloud may hit any part of Jenova-SYNTHESIS since he is so weak that his damage does not matter in the least, and Tifa's final 2xCut(s) may hit Jenova-SYNTHESIS B and C as well without losing too much.
I was quite lucky that Tifa's third 2xCut hit Jenova-SYNTHESIS A instead of her arms, resulting in a perfet battle. Still, even in the worst case, Tifa killing both of Jenova's arms first, I would have lost only 10 seconds.
Bizarro-Sephiroth is the second final boss and by far the easiest, thanks to ATB on Wait.
There is only condition that must be met: His opening move, Sephiroth Shock must not kill anyone or be critical on Cid.
One word of advice: Do not even consider killing off Bizarro's other limbs, for you lose more than 15 seconds on waiting for a "Sephiroth's ugly and feeble body part ... has died" notification.
As Bizarro has solely 40,000 HP, Cid and a heroic Tifa will Bizarro in three turns.
Safer-Sephiroth is the truly final boss and ranges on the difficulty scale somewhere between Bizarro-Sephiroth and Jenova-SYNTHESIS.
A few things must work out in this fight: Cid must not miss any of his two Deathblows, Shadow Flare may not kill Tifa or Cid and Safer's physical attack may not kill Tifa or Cid either.
To be perfectly honest, Cloud's only purpose in this battle, after having cast DeBarrier on Safer-Sephiroth, is that of a human shield. His only considerable source of damage, Ultima, takes too much time to perform and barely deals 6K HP of damage.
Finally, there is Sephiroth and I found a revolutionarily entertaining new way of dealing with him... ;]

Thanks to everybody who has watched my speedrun. I hope you enjoyed it.
Special thanks to:

-GarlandTheGreat for being great and having written a speedrun guide I made thorough use of
-BrutalAl for being brutal and having provided me with several ways to gain time through his glitch discoveries
-Gjørulv for being handsome and having reincited my interest in FF7 in 2010 with his Hardcore Mod
-Acro of ff7.fr for being awesome and having created the greatest FF7 website I know of