This looks brilliant already. The little feather / ear bounce is particularly charming.
I'm no expert on anatomy, but it seems to me that his left leg is pointing so much to our right that it would be impossible for him to bend the knee without moving his knee forward. But maybe the slight bounce is not enough to warrant the knee moving a full pixel to the right, I don't know.
I don't know how people tend to do idle animations, but my impression is that they're meant to look like heavy breathing and/or idle movements like stretching, flexing, etc. This guy seems to have a lot of vertical movement from the bending of the knees, while his arms are completely locked and are barely moving independently of his torso. It actually makes him look a bit like a boxer, who's keeping his arms tight on purpose (as a guard), while bouncing up and down. But then, you would expect him to stand on the balls of his feet. If this is for a game that is primarily about running and jumping instead of fighting, then perhaps you could loosen up his arm, shoulders and torso, like he's taking deep breaths. Hell, even in Street Fighter 2, many characters look like they're taking very deep breaths when idle.
In regards to the running animation, I don't quite understand what you want to know. Are you simply not sure how to start drawing a basic running animation? If that's the case, then just googling running animation is always a good first step. Beyond that, I'll leave any elaborate explanation to more educated artists. I've seen a lot of people say that it's very hard to do a nice running animation with only 4 frames though, so you may want to reconsider and go with 6 or 8.
Hope this helps, even though I don't really know much about animation. People with more knowledge will come along shortly, I'm sure.
