Just a few things to add to the good comments you have already received.
Higher ISO levels actually reduce contrast. ISO 500 should not be a problem, but lower is better.
1/1000 sec shutter speed is marginal for birds. A little faster is always nice.
Assess the lens and practice under easier conditions - bright light on the subject, stable base, and close range.
The D7000 has nearly the same pixel pitch as the D800 - so a 100% crop is a very small section of the image. But at the range of this bird, you should be able to get a sharp image.
AF-A sounds like a good idea, but at the instant the subject moves from stationary to flight, the camera seems to stay in AF-S and fail to fire. I found the instant of motion to be consistently missed with the camera failing to fire. I only use AF-C for wildlife.