Oh man, I love Super Mario Bros 2! It's one of my favorite games of all time.
So, each character actually has very different stats.
Luigi and Princess have obvious special jumping abilities. I think Luigi can actually jump a little farther... But both of them have significantly shorter jumps and run more slowly when carrying a vegetable, and both pick up vegetables quite slowly (especially the princess). This can be extremely detrimental on sand stages that require a lot of fast digging to get away from enemies, or picking up a block to throw at a boss before he attacks.
Toad is the opposite. He can pick up vegetables and run with them blazingly fast, and can run and jump as high (higher?) while carrying them.
Mario is balanced all-around.
It's all explained in the instruction manual with some cute drawings. Each character is really quite balanced in my opinion.
In the US version of Mario 2, all you have to do is make it to the end with any character to win. In the Japanese Doki Doki Panic on which it is based, you actually need to get to the end with all 4 characters before you've beaten the game.
The different characters do have strengths and weaknesses: differences in run speed, item pickup speed, and jump mechanics (and Toad actually runs faster when he's holding an item) but you are correct in that for purposes simply amounting to no more than getting through each level to the end, using Peach/Toadstool is the easiest.
I think you could recreate that experience a bit of you go after the retroachevments for the game (e.g. fan made achievements enforced via emulator). I think they require you beat the game 5 times, one for each character and once with a split group. Also, no damage runs and such. Difficult stuff.
That's always been my feeling too. I think Mario can run faster, and Luigi can jump a little higher, but Peach just controls better. Who knows what kind of maniac would choose Toad.
There's a warp zone you can only reach with Luigi, IIRC (but it's been a long time). So for one level, at least, he's better if you want to skip ahead.
Yeah, they're pipes / jars / whatever. You have to find a potion, throw it so the pipe is available when you go through the door, and then go down the pipe.
Peach was basically my only character ever, but occasionally
I would use Toad for the Ice level with all the whales since he could jump between the bees easily.
If you want an overly objective answer, speedrunners use different characters for different levels because they're faster in different ways, and I don't think Peach is used often. As a casual player IDK lol I always just played as Peach
Speedruns can be a lot of fun to watch - the better you know the game, the more fun it can be. Some runs get freaky, but a lot of games have glitchless categories or some kind of ruleset that doesn't break the game too badly. A lot of glitches are fascinating, too.