Cool idea, and an original combination of things for the three buttons to do! You made some good puzzles with these mechanics, and the later ones definitely made me think. The player's move speed was definitely right for the later levels but a bit slow on some parts - I would say you could add a button to move faster, but that would break the restriction :) Maybe hold z to run? In any case, great work!
I hadn't heard of the website you got your sound effects from, I'll have to check them out.