FX LENS ON DX BODIES?

Of course you can.

The problem lies with the camera body, the D3xxx, D5xxx and some older models don't have the body motors that would allow AF-fit lenses to autofocus, not a problem with the D7100.

DX lenses take advantage of the smaller sensor to make the lenses smaller and lighter, but FX lenses are perfectly compatible.

Yes. No problem.

Not only are FX lenses compatible with the D7100, because they are designed for a larger sensor you shouldn't experience any sort of vignetting which you might do with a cheap DX compatible lens at its widest setting.