OK, here's my WIP up to the fadeout of the second level.
Improvements:
-Managed to skip the second story sequence one frame faster.
-Started running one frame faster in the second level (The point I used to measure this was the first frame the character moves two units forward instead of one)
-This level in some ways didn't seem to have as much to optimize as level 1: no flying sequences, no trying to go over a lake or up several ledges. There was one thing, however -- since this level has a multitude of places where you have to jump, various ways to move through the level thanks to the branches, and of course, enemies to stand in your way as you leap through the branches, I was able to save a decent amount of time as a result of attempting to find a route that minimized the number of jumps and ground attacks, which cause small delays in forward movement. Much messing around with that... I'm certainly much more familiar with this level now. I was able to save 22 frames in the main portion of the level measuring up to the semiarbitrarily chosen X position indicated by 7E05AF=2512 (appears to be for a scrolling layer, but good enough...) through this kind of optimization, and I was able to fit in some air attacks on enemies... if done right, those don't cause any delay.
-The last section of the level was interesting to try to optimize. Measuring from after you jump over the pit between two branches, it seems like getting to the boss takes a bare minimum of three jumps. I was able to find a route through that section with only those three required jumps with no attacks whatsoever, yet this didn't end up giving me the best time I could manage when experimenting with various ways to go through this section. My times seemed strangely incosistent with what I know of how the character's movement works. My guess is that it has to do with lag. At one point in this section, there are four enemies on screen, assuming you don't kill them. My best attempt had a one frame delay apart from the neccessary jump delays, due to a fireball hitting a pirate. I'm guessing the reason that attempt worked best anyways is because it kept the number of enemies onscreen down. If it was indeed lag, I suppose I'll have to watch out for lag in the future when I notice lots of onscreen enemies. BTW, I didn't notice strange behavior like this in stage 1, probably because there is, in my run, just one brief point there with three alive enemies onscreen.
-Exact same number of frames per hit on this boss as in hero of the day's run. However, I took advantage of the acorns to minimize my life to use the score tallying trick. Yes, that means I lose the Golden Sword, although there are places to get the sword back in later stages. Hopefully, losing the Golden Sword I had in level 2 won't make me lose frames later, but if it appears later that I should have kept the Golden Sword I had here, I'll go back to this fight. Anyways, I also managed to, on the last hit, position myself so that I was next to the southeast hole, and yet not get hit by the boss emerging, or by falling acorns. So I was able to kill the boss and also be very close to the bottom-right edge of the screen. I had Peter Pan already running once the screen starts to scroll right. I saved 11 frames in getting to the point where Peter Pan gets into his end-of-the-level pose. Quite frankly, I'm not sure exactly why, because while hero of the day did exit the boss section in a different way, it looks like he was running at the bottom right of the screen when the screen started scrolling to the right.
-32 frames saved in the score tallying screen, if I'm counting correctly.
-In total, 84 frames saved measuring from level 1 fadeout to level 2 fadeout.
Level 2 WIP