Cohesion. I mean that in the most general sense, applicable to whatever style a person is going for. That means if you're going for color matching, all your shades are spot on and balanced; clutter, your texture runs through the whole thing; themed, all your items lend to the character/atmosphere/theme; etc.
If an avatar is not cohesive enough, it will fail, regardless of how well other elements might be working individually.
As for what's least important, I'm not actually sure I can answer that. You're right that it's a more difficult question. I'm not sure I can give a generalized answer like for the most important. I think what's least important might actually depend more on the specific avatar. It's whatever detail means the least to the overall affect. So, for a themed avatar, getting a really specific shade to match another may be the least important, whereas with straight up matching, it's totally okay to put a random thing on your avatars head for color balance reasons.