yes. but it'll be "cropped." i've used a nikkor FX wide-angled lens on my d7000 and it works but i didn't get the full benefit of its widest mm because it was cropped to fit DX. sorry for the very simplistic, layman terms. not a pro. just love photography and i own a d7000. i've rented both FX and DX lenses for the d7000.
yes, but there will be a crop factor. So 35mm will be closer to 50mm, 50 closer to 80mm, 85mm more like 105mm. You might experience a problem with wide-angle lenses (12-24 etc) where you will see black. Hope this helps
I know many would tell you not to buy FX lens because of the crop factor, but I would tell you that it is the best thing you can do buy the best glass. I have the D-7000 and all I use is the FX lens in fact I sold the DX lens. One more reason to buy the FX lens, if you upgrade to the FX format the DX lens will make it shoot in DX with the disadvantage of reduced angle and with lower MP. Good luck