In the first shot - you can blur the cage wire so that it ALMOST doesn't show when the bird is far away from the wire by placing your lens very close to the wire and shooting with a wide aperture to blur both the foreground and background. You may have used that technique in this shot, but the bird may have been too close to the wire. I'd like to avoid these shots also, but sometimes it's worth trying to get a shot of a bird/mammal that is interesting to us - or for practice.
The third shot looks (to me) like an attempt to capture those lovely wings, but a much wider shot would have improved the image. I certainly agree that focusing on the eye is, in general, very important to capturing the beauty and spirit of any living creature - human or animal. Moving to another position is a very important part of photography -- we need to consider as many views as possible - sometimes, all we can get is a quick grab shot, but often it is up to us to find a better vantage point.
The second shot is sharp on the subject, separating it from the background. Again, walking to each side and searching the viewfinder can help us find the best available background - which sometimes still needs to be blurred on purpose.
You're going to have a lot of fun with your new camera set-up! Keep shooting!