You can remove the small twig either by cloning it out or doing a content aware option. The eagle looks nice despite your critique that the wind blows. Hey! the weather comes into play, you can't comb its feather just to make him look good. Consider it your very own style.
I kinda like your shot of #1 but the bird is a little bit on the soft side. You are right -- it would have been better with a faster shutter speed. #2will be OK if you lighten up the exposure in your PP and put a bit of contrast so she will stand up to the test. In short, excellent series.