I have not seen a full game run of SMB2 so I decided to take on the challenge. I used the same tactics for original stages as previous authors and then used some more of my own.
Emulator: FCE Ultra
Objectives: Completes game as fast as possible Takes damage to save time Does not use warps or passwords Abuses programming errors Manipulates luck
Stages are as follows. 1-1: I use the same technique as Sleepz because it is what seems to be the fastest.
1-2: Again, thank you Sleepz for showing a faster way, I also do the same.
1-3: I use Luigi because I could not figure out how to get the star perfectly to kill Mouser. So Luigi became the next best with jumps. And the UP+DOWN is first used big on this stage. Mouser however is beaten very fast I think.
2-1: Pretty much just a run through the stage until I hit the sand digging. From there I have to get some cherries in order to get my star which for some reason(for me) would only appear after the enemies projectile hit the wall and went a certain distance off the screen, so it didnt matter how fast I dug, also I had to press start 3 times to make the game think the star went to the left and then it gets a more needed early start across the screen, from there I have fun with the Birdo.
2:2: Again, running to the end and then digging my way down.
2:3: And again, running and digging. But when it comes to the boss, I use his little buddy against him to kill him faster.
3:1: This stage is a little different, more of a go up stage rather than a scroller. Nothing special really here cept for the Birdo beatin.
3:2: I chose Princess here because she can float for a good amount of time. You will see why. It just seemed the shorter route. Anyways, Birdo gets taken care of nicely.
3:3: Luigi was used for a particular reason in the level. His jumps are just the right height. Mouser gets destroyed at the with every bomb. Fastest possible way I thought of destroying him.
4:1: Same as previous speedruns.
4:2: Same as previous runs, until I take the rocket up to the next level, then I get enough cherries to make the star appear to destroy Birdo once again with the touch of my body.
4:3: Luigi worked perfect for this stage. His long jumps and his better moving ability when he picks up objects(I think). Plus, the boss gets it.
5:1: Good timing with Luigi is needed on this, plus I use the (Pick up item after returning) glitch to take care of Birdo faster than normal.
5:2: I first thought Luigi would be good for this stage, but Toad was the better choice with the faster running and all. I fish slap the Birdo at the end :)
5:3: Pretty much a running through the level. I used a faster technique than what I normally planned for Birdo. And the crab-looking-boss at the end of the stage is easier than holding A.
6:1-7:2: These stages are pretty much the same as the speedruns. I dont want to explain them because the previous authors have aleady.
ENJOY.

Bisqwit: You cancelled this submission, which was appropriate given that you're already improving it, but I'll publish it anyway. :) I hope we'll see the improved version too in the near future.