Handled well, Nice engine, Very few problems
Give yourself a hand, you have created a great game.
Now for a better game, implement a lives system, save points, more levels, longer levels, a better fighting system, a little arrow pointing us towards whatever gate was unlocked so we don't have to sit around and wonder where said gate is.
Not to mention, the beat the daylights out of the enemies can not only get tiring, but it becomes dangerous. Wait too long and you're swarmed and unable to continue. (Mainly if you have something like the jump hat ability, since your offensive abilities becomes that punch) Instead of randomly dropping enemies, place them in a set location and when we need the ability we would either grab it from a brawl, or go back and get the ability.
Overall, the game feels nice, plays like a dream, and is one of the better games I've played on Newgrounds.