I don't exactly understand what you want clarified about TAS. But it usually works like, a player goes into an emulator and slows the game down so that they can advance frame by frame and have as much time as they want to make a decision. Once they've decided what they want to do, they go to the next frame and then decide for that frame. There are 60 frames in a second. Usually when making a TAS, they make savestates so they can also go back and redo a decision they don't like. If two opponents are battling in a TAS, they have infinite time to make the decisions they want each frame, though they probably can't take decisions back.
Like I said at the end of my post, how good a character is at the TAS level does not reflect how good a character is at a top theoretical level of play for real people. There are three big distinctions between TAS and normal gameplay. One, TAS has infinite time to make the perfect decision. Two, TAS can be perfectly precise everytime in the decision they make. Some techniques require very exact timings or controller inputs which TAS can get everytime. Three, TAS has an inhuman reaction time. They can see and react to what happened instantly. Whereas for a human, it takes about 18 frames to differentiate what just happened on screen. As you can tell these are some really big differences between optimal human play and TAS play lol. For instance at TAS play grabs have little use for most characters. Why? Because spot dodge comesout frame 2 and grab comes out frame 7. So at TAS, your opponent tries to grab you, you react frame 1, spot dodge, and punish the grab. So Marth has no very strong strengths at a TAS level. This obviously is different than a peak human level because humans can't react to grab coming out and dodge in time. TAS values different character strengths than top theoretical human play. For instance Mr. Game and Watch has a move that comes out frame 1 (up b). That alone makes him better than most of the cast at a TAS level, since frame 1 moves are the only moves that can't be reacted to. Similarly Bowser has an up b invincible frame 1, so he can only be attacked by frame 1 moves safely at TAS. These characters being stronger than most of the cast at TAS levels does not mean they are on top at human levels. Hopefully that makes sense now. Just because Marth is mediocre in TAS doesn't mean he's a bad character.
It sounds more like you're wondering how good Puff is at a top level of human play, where is Puff on a modern tierlist. This is really hard to say and a lot of people debate the topic. I'll say that at worst, if you judge Puff as harshly as possible, she's the fourth best character in the game. Armada and Leffen have Puff as the best character. I think Mew2King and Plup have her as 2nd best though I'm not sure. Hungrybox thinks she is the 4th best character. A tier list was voted on collectively by Reddit recently and Puff was 3rd best. There's a lot of debate and it's hard to say. That said, it can be said that Puff is absolutely a top tier in Melee. Like she is definitely the absolute top of the pack. She's probably a top 3 character, in a tier apart with fox, marth, and puff in some order. That's not a very definitive answer but no one knows for sure how good Puff is. She's really good though lol.
I'll speculate super quickly about the matchups you mentioned on a TAS level:
Falco > Puff
Sheik = Puff ? Puff could maybe win
Captain Falcon = Puff
Marth = Puff ? Puff could maybe win
Pikachu = Puff
There can be ties at TAS level if neither player can ever hit each other.
Hopefully this was helpful and cleared some things up.