Wow....according to the gamefaqs.com guide, I am 20 minutes ahead of schedule. Thanks for pointing that out to me. 
Now, before I start with the Iifa Tree, let us once and for all end this silly argument about how this location is spelt.

Now, everyone sees the upper case "I", and and everyone also sees the upper case "L". Let the arguments end. And now....
Iifa Tree
-You know, this used to be my favorite place in the whole game. Even when doing an Excalibur II challenge, it was my favorite place, we could get into like 12 random battles here and be fine on time, no worries. But now....not only do I not have that much time, but I'm also at level 1. Stroper=death most of the time if it gets an attack. Also, all of the winding back and forth of the path in this area is just asking for random battles, which is not good.
-But first, I just have to get to the moogle and save. Some well placed stop-run action gets me through all but the last screen without a random battle, the last screen in which I got one. Before I save, Zidane gets the Ritual Hat to bring his magic defense to 19 and his evade to 17. Dagger and Eiko both get the Moonstone (since I have 2
) so they each have Shell. I also deliver a letter.

-Alright, buckle down for this long segment. Every guide and video that I have seen tells you to go down, fight a Dracozombie and get Zidane zombied, come back up and save and go back down again. I don't like this setup. I think it's much faster to just go down and start treasure hunting. The random battle rate is kind of high, and the guides alot for 8 random battles. That's really good odds for fighting a Dracozombie, so why not kill two birds with one stone? I understand the logic behind the above setup, I just don't like it.
-Since I'm not coming back up and going back down, I can either save about a minute or I can get into a few more random battles. Either way, it will make this section easier, which is good because I want out of here as soon as possible. So I go down the elevator and equip Zidane with the Mythril Dagger and Bandit (which I should have done with the moogle before I saved, but oh well).
-The second screen has a switch that will lower a chest. Sometime in this screen I get a fight with a Dracozombie. With Zidane's current setup, Shell status and full HP, if he's hit by Zombie Breath he'll be left with either 9, 21 or 33 HP. I don't want 21 because that means I have to use a Sapphire to make his HP end in 7, and I only have one which I need for later. The other 2 will work though. Zidane steals a Magic Tag, which is what I wanted, and Vivi uses a Tent successfully. And crap hits the fan as Dagger gets zombied. So Eiko self-KO's, Vivi uses a Magic Tag and Zidane revives Dagger. Vivi self-KO's, Dagger uses Shell on Zidane, and Zidane KO's Dagger. Shortly after, the Dracozombie uses zombie breath again and Zidane gets zombied and goes down to 33 HP, which is exactly what I wanted. Then an elixir ends the battle.
-I go down to the next screen, flip the second switch, and go back up a screen. I get a Hi-Potion, Phoenix Down and Healing Rod. I go back down to the next screen and get an Ether, a Lamia's Flute and Remedy, and then go down to the leaf. All in all, I think I got 6-7 random battle excluding the Dracozombie fight, several of which Zidane started with nearly full ATB. Not too shabby. During one of these battles, I had someone use an Ore on Zidane. Since I had 11 Ores, it gave him 44 HP for a total of 77. (And if you guys go back and watch 1whoistornapart's videos, he skipped the Ether. I wonder if he knows about that)
-And now for some forced battles. The only thing to worry about here is Zidane getting hit. As long as that doesn't happen, everything should be fine. The first battle I only had to fight 2 Zombies, which made things a bit easier. One Zombie went down with an Elixir, and then I waited for the other to make it's move. Eiko got KOed, Dagger and Vivi self-KOed and Zidane used an Elixir to win the battle. After some dialogue Zidane says "Vivi...", and that's my cue to revive everyone. Then a Dracozombie battle. I waited for it to make a move (and I think it started with an empty ATB or something). It zomibed Eiko (poor Eiko in this segment), so Dagger and Vivi self-KOed and Zidane used an Elixir for the win. And everyone is still at level 1.
-After getting to the final area and some dialogue I get the Elixir and Brigandine. I use a Magic Tag on Zidane and Eiko and revive everyone. I equip Zidane with the Brigandine, Dagger with the Magician Shoes, Vivi with the Lightning Staff, Magus Hat and Peridot, and Eiko with the Lamia's Flute and Ruby. Zidane gets Ability Up.
-And finally, for the easiest boss in the game, Soul Cage. I either would have used Steal with Zidane and someone else throw an Elixir, in an attempt to steal the Brigandine, or just have someone throw an Elixir, depending on how everyone's ATB started. Getting this far with everything going well is no small feat, so this is one of those things where however this goes just take it. Dagger started with nearly full ATB while everyone else's was almost empty, so she threw an Elixir and won. Zidane learns Bug Killer and Distract. Dagger learns Ability Up, Silence, Locomotion and Blind. Vivi learns Slow and Ability Up. Eiko learns Float.
-After skipping an FMV, some dialogue and exiting the Iifa Tree, I take a few steps to turn the camera around and save. I ended up taking the "save about a minute" option.

Bounty Hunters, Round 2
-I didn't take that many steps out toward the Mountain Path, so this took several resets to not get a random battle. And then a few more not to get a random battle in the Mountain Path, but just a few on that one.
-When I get to the very end of the Mountain Path, I change some equipment around. Zidane gets the Gladius, Bandana and Chain Plate. Vivi gets the Mythril Vest. Eiko gets the Bandana and Silk Robe. Zidane equips Distract and Man Eater and Eiko equips MP +10%, Ability Up and Insomniac. This is my last chance to equip Eiko until she forceably joins the party in the Iifa Tree, which is why I'm re-equipping her here. I save half-way to Madain Sari.

-And now for some more Madain Sari fun. Except for the Scarlet Hair battle, this is just a bunch of dialogue. Dialogue, dialogue, dialogue. The only hard part of this is another Lucky Seven battle. And that is after half of this dialogue. The fingers get a little tired after several retries.
-So the Scarlet Hair battle could go in one of two ways to satisfy me. 1: 2 Lucky Sevens doing 7777 each. 2: 3 Lucky Sevens, 2 doing 777 and 1 doing 7777 damage. The first one is asking a lot, but it is possible, and the second one would only require one more round, which isn't too much more time at all. With Zidane's setup, he has 19 evade, which seems to work rather well against Amarant. After several retries, I got the latter, which I'm happy with. Zidane learns Devil Killer.
-After the rest of the dialogue, I name Amarant, switch him and Eiko in my party, go half-way to the Mountain Path and save.

Battle of the Bad Guys
-So now I make my way back through the Mountain Path (for the last time) and head to the Iifa Tree (for the last time). I equip everyone as follows: Zidane: Mythril Dagger, Ritual Hat, Mythril Armlet, Brigandine, Reflect Ring
Dagger: Multina Racket, nothing, nothing, Silk Shirt, Moonstone
Vivi: Lightning Staff, Bandana, Leather Wrist, Survival Vest, Fairy Earrings
Amarant: Cat's Claws, Ritual Hat, Mythril Armlet, Adaman Vest, Germanis Boots
Zidane gets Distract, Ability Up, Insomniac and Bandit. Dagger and Vivi get Ability Up and Insomniac. Amarant gets HP +10% and Alert. Vivi's equipment is completely different from what the guide says, but this is as high as I could pump Vivi's Spirit, which is what I was going for.
-I save half-way to the Iifa Tree.

-This next section is very annoying. First, I need to get through the Iifa Tree roots as quickly as possible. And then I must rely on Stop 3 consecutive times. The higher in level these monsters get, the harder it is for Stop to work. Not to mention all of the dialogue.
-So, after many many many tries, I decided that it's ok if I get a random battle while going through the roots. This let me try the Mistodon battles about five times as often.
-The save that I kept, Zidane used Flee almost immediately during the random battle.
-During the Mistodon x2 battle, 2 Tents were used, both of which worked. And those were followed by 2 Stops, both of which worked. Zidane learns Bright Eyes.
-Then with the single Mistodon battle, Vivi got the first turn and Stop worked. Eiko learns Antibody.
-It took me a long time to get this save, and it was worth it. The only way this section could have gone better is without the random battle, but it was a fast random battle so very little time lost. Eiko gets the Magic Armlet, I read a letter from Stiltzkin and save.

-Since this next section is so close to a moogle, it wouldn't be too difficult to do it perfectly. The only hard part about this segment is making it to Dagger with just one battle with 1 Mistodon.
-After many tries, I got a battle where Vivi got the first move and Stop worked. Vivi learns Thundara and Eiko learns Insomniac.
-I make my way down to Dagger, go through all of the dialogue and FMVs, and save as disc 2 finally comes to an end.