Yeah, I think timing and rhythm is very important!

Get some key actions, and play with the space in between! Those actions don't need to take up 20 frames, 3 can be enough, save you work and improve the result

I did a very rushed attempt to give the whole thing some more sense of rhythm, but it didn't turn out very good. I hope it can help anyway. Pay attention to all the moments where nothing happens, and how little frames are needed to whip out the gun! My lower framerate is only noticeable in the gun charge, which is a very linear thing.




Have fun with it

Laser is just light, why give it a dark outline?
Also, I think in your animations you try to smooth things out too much. The building up of the whole gun could perhaps be funnier if it appeared more clunky...
Same goes actually for the flash effect on his body. With a one or two frame full on flash you might achieve a better effect than this soft brightness fade..

If you don't let it loop so fast, you'll notice that the arm goes back really quickly, I guess it could stay there longer.
Why does the beam appear so softly? Make it come in with a bang! More surprising, and feels stronger! No need to draw those buildup frames.
My quick edit:



Good luck!
 

Not bad! But I've found some issues 

Where does his hair go during the roll?

When you are exactly vertical on the moment of landing, you'll smash your back into the ground very painfully, no way to turn that into a roll unless you have a lot of vertical speed. This guy has horizontal speed, he would need to enter the floor at 45 degrees or lower!

The first frame seems a bit useless, I thought I'd replace it with some anticipation. After the explosive moment of release, the last contact frame, he would be far away from the ground. 1px from the ground is never a good start of an animation!

I turned the ball a bit more in a ball, and I stretched him out a bit during flight. Also I made his landing a bit more smooth and realistic.



I hope my edit makes him look less like tumbleweed, and more like an action hero!
Good luck with your progress

6 frame run cycles are hard! But not impossible.
I recommend you use this image as a guide, only look at the odd numbered frames


The important moments are:
The first connection with the ground, this one is kinda present in yours but not very extreme
The release, the last connection with the ground. This one is not to be found in your animation
The jump position. The odd numbers from this guide would suggest that it is also possible to leave it out, and go with a mid contact position instead, but I'm not sure how well that would work with 6 frames.


I tried to do a quick fix on your run, but there is little I could do quickly without redrawing the whole thing from scratch

Welcome to the wonderful world of animation 

The up and down motion seems to be out of sync with the wingflapping.
Wings flapping down = driving the body upwards. At all other times there is a constant gravity pull, and perhaps even an extra downwards drive when the wings are going back up.
I tried to fix that a bit, but I'm sure it could be done even better:


Also there is some funny stuff going around in the head and the neck. It is nice for the head to move separately, but that the size/height of the head and neck change is a bit strange...

As for bat anatomy: he seems to have two wrists! remove the one on the far end! Also his wrist would be more extended in the downflapping part, and his elbow more bend during the upwards recoil.
A quick google led me to this ref pic, which seems fairly accurate to me. That page also links to one of my favorite animation articles of all time: How to make birds fly good

Good luck with your animation progress!