Hey, I played your demo! Here's some feedback, as requested.
The animations and art feel pretty good. If you can get the combat and controls up to snuff, I think things will really shine.
Double jump should have an animation attached to it and a different feel. I recommend making the wings thing the double jump instead of having it separate. Then right click can be the super blast instead of alt. I think this would feel more natural and more interesting - it was hard to get used to hitting left mouse or alt to use weapons, hard to remember to use arrows and right mouse to dash.
Up shouldn't jump. 99% of action games use up for interact or pick up, instead, and there's a reason for that.
When jumping and hitting a ceiling, consider having the character either bounce down or grab hold - just floating along feels weird and weak. This may be less of an issue if the second jump is that jet-assisted wing thing.
Weapons: a gun which pushes you a little bit feels waaaaaay better. Try it, you'll see. Also, in these non-ortho-gunfire games, you should make a circular impact sprite and use masking to put it behind solid objects. Otherwise it 'floats' off the walls when you shoot them at an angle.
Basic running: consider a somewhat fast initial burst followed by a more steady run. This is ideal for games where dodging is a consideration.
Enemies: bullet sponges are boring. In general: more weaker enemies, or enemies which strongly react to getting shot. For example, robots could lose their arms and then dash at you in a self-destruct mode. If an enemy takes more than five shots, you'll probably want to rethink it, because it stops feeling satisfying.
Camera: screen shake is good, nice. Lots of screen space wasted on areas outside of the level, though: may want to constrain the camera to level bounds, if you can.
You'll want to recapture mouse constraints after losing focus, or your game will be unplayable if anyone checks twitter while playing.
Anyway, good luck!