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(+1)

About the jump: Agreed, i'll remove the jump delay, maybe the game even feels smoother without it!

About lives: Here'sAndyTV also addressed these issues, and suggested a very good approach for the health system, which is: when get hit, just lose a heart without returning to the begining, but after losing all hearts, there is a number of main lives that decrease and then you returning to the begining of the level. Obviously, after all main lives are gone, its game over. Sounds way better, doesn't it?

(+1)

Yes that's sounds much better and more enjoyable and forgiving for the player. I would done something similar in my game but being a platformer that takes you back to the beginning of the stated level their no real need because they also rather short.