Hey! Look, more time for the forums 
Yay!
I totally love the character design, reads so clear that is scary, I may have to put you down to eliminate the competition 
Haha, thank you so mu-- Wait, maybe I shouldn't be thanking y-
First of all I have to say I've never worked with restrictions in mind so my advice may not be as useful for this specific case but here I go:
- As it is, well, the punch feels weak and the reason is exactly what Cels told you, It's all about those hips baby, my advice when animating anything is to picture yourself as the subject, in this case, throw some punches and try to feel what every part of your body is doing.
- More specific, I think there are two principles of animation that can be applied here without hindering playability: Anticipation and Overshoot. Since this is for a game anticipation can't be actual frames, instead of that use the smear to convey that anticipation, simply draw the smear from his back to the front, this gives the impression of anticipation. As for overshoot, well, in this case is better to add another frame for it and push everything a pixel further, the feel a single frame can provide is out of this world.
- Again, not sure of the palette or the restrictions, but for the idle you may want to try sub pixel animation like in Metroid for the SNES, so, some movement of the outline, followed by movement on the shading only (Pretty much an after image, like Megaman dash)

I'm sure my edit has many improvements that can be implemented to give it a better feeling so if you guys have any ideas I will be really happy to read them! I hope the edit helps with your project!
Dude. Dude!
Let me start by saying that that animation blew my mind so fricking hard I had to take some time to recover. xD
That's awesome, I love it, and will be studying the heck out of it.
That's a beautiful lunge (if that's the right word..?). I tried something like that in my first attempt, but then I thought it would look wrong making the body go forward too much. Also I'm trying kinda sort of to be somewhat frugal with how many (8x8) sprites I use So I may reuse some parts here and there to accomodate as many move sets as I can (attacking, jumping, rolling, etc.) and because of that I maaay not be able to add as many frames of animation as I wanted. But your animation is making me reconsider that. xD
I need to start planning things. Write down some stuff and define most of what I want the project to be...
Anyways, for an NES game (which is what I intend this to be, and not just a modern game that would just look like one) I have a limited amount of space for sprites (though not sure how much yet), a limited amount of sprites on screen at the same time, a limited amount of sprites per scanline (8, or 64 pixels) and can only use 3 colors per sprite, from 4 sets of 3 colors each.
Thus I don't know if I could use subpixel animation, at least not without stacking sprites on top of one another, but I want to avoid that as it would limit what I could put in front of the character without making the game turn into a flicker fest.
I mean, I know there will be some flicker, that's practically unavoidable if my main character is 4 sprites wide and there are two 3 sprite wide enemies in front of him. But... we'll see.
Still, it was so great to see your animation and it has definitely made me see things in a different way. Thank you so, so much.
@cels Thank you. The edit on the foot looks great. I was afraid it could make the stance look a bit weak, but you're absolutely right with that. And with the body/hip twist too, of course.
The twist on mine was a bit of a last minute thing before posting, because like I said I'm trying to be economical and reuse stuff, but I can most definitely work with that.
Next I think I'm going to do a walk cycle. Or a weapon attack.
Again, thanks to the both of you. I was super happy when I saw the replies. Cheers!
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EDIT: Oh, if you guys are curious, there's a bunch of talk about the NES limitations on the pinned Castlevania Pixel Quest thread.