Just a minor clarification. The D600 uses the CAM 4800 AF system. The D7000 uses the CAM 4800 DX AF system - a different layout optimized for DX. The D800 and D4 use the CAM 3500 AF system. The D7100 uses the CAM 3500 DX AF system.
There is a much greater drop off in the DX version of these AF systems. For example, the D7100 has only one reliable AF sensor beyond f/5.6. The layout is not disclosed on the D7000, but I would assume it to be very similar to the D7100 system. http://imaging.nikon.com/lineup/dslr/d7100/features01.htm
This issue mainly comes into play with lenses or lens/teleconverter combinations that go beyond f/5.6 wide open. That's not the case with any of the Nikon lenses until you add a teleconverter. But the long zooms from Sigma - the 50-500 and 150-500 - have only one reliable sensor at the long end.
By reliable, I am referring to the fact that in great light with a good AF target you may be able to get AF to work, but it is significantly less accurate and will be more likely to hunt or give up.