1. The up punch has more than enough to reach the 1st lair of enemies, but if you're having trouble I'd recommend pressing the jump button first.
2. The punch button is 1, because there's 4 other punch abilities. It wouldn't make sense for the 1st punch to be mouse and the others to be the number keys or some other variation. It's the control scheme that makes the most sense.
3. The player takes damage as a punishment for mistiming punches, additionally if you get high enough on the tower there are enemies that will actively chase the player and try to deal damage to them, as well as the boss fight. If I took your suggestion and made it so the player didn't take damage then I'd have to remove several enemy types.
4. I understand there isn't a lot of hand holding in this game, but at the bottom there's the 1st upgrade that unlocks the up punch, then the 2nd upgrade looks exactly the same except the arrow is rotated slightly. The idea is most people should be able to realize it's an upgrade by looking at it, but if enough people have this problem I'll definitely add some extra instruction and maybe a tutorial.
A final point, the game is fully playable in the current state. I tested it and was able to get all the way to the top. Granted it took over an hour and I understand you probably don't have that much time and might not usually play games like this. As far as recover, the very next upgrade you would've unlocked was the down punch upgrade and that's effectively what it is.