First with the non responsive jumping i mean by when i jump and land i cant jump again for second or 2 now that would not be so bad if the enemies weren't a tad faster then you or if you weren't cornered as it when i got cornered i would be mashing the jump button hoping the key press would resister in time.
for the type of game you looking at developing if you fell a bit faster after a jump it would be easier for you to then jump again to say jump on enemies or get away from them. because at the moment i find it difficult when i miss jump and their enemy almost under me that i hit and die on.
in terms of health the heart system is good but adding a health then losing a heart would be better suited for this type of game. The other way you could do is just by decreasing the speed of the enemies slightly or adjusting the jump mechanic slightly