Dani, what is your question? If it is how does this lens perform on a D600, the answer is quite well, and the lack of sharpness at 300mm reported on DX cameras is not my experience with the D600. Because the D600 does so well in high ISO (low light) situations, you would be pleased with this lens when your meter says you should have an aperture wider than 5.6/4.5. This is an excellent travel lens, hugely lighter and smaller than a 70-200mm 2.8 VR.
Yes. It will fit, it will focus, it will work fine on the D5000 just like on the D7000. But remember that it is a lens designed for a full sized sensor so you will have a 1.5x crop factor, just like on the D7000. Personally, I don't find that to be an issue with telephoto lenses.