Pretty good!
Few comments, take them as you will.
The spikes, maybe have the tips shaded a combo of gray/black so it is more obvious they will kill you.
The wall stick feels weird, maybe have it so you slide down the wall till you jump again?
If you fall a long ways, have you die, hence requiring jumping down walls to get down to the bottom of a screen.
Otherwise, like the atmosphere and general movement, gives me a slight Super Meat Boy vibe.