Are lenses interchangeable from photo to film?

Are lenses interchangeable from DSLR photography cameras to DSLR filming cameras from the same manufacture?

E.g. Can you use a lens used on Nikon D200 (photography) on a Nikon D600 (for filming)?

I have a 50mm lens I want to use from filming.

Yeah. Everything should work.

Yes, you can use same lenses for photography or for filming.
However if you want to get more in filming, it helps to use faster lenses for getting better control on the depth of field. Also, it helps if you can control aperture on the lense itself to control depth of field.
Your 50mm lens, I'm guessing must be 2, 1.8 or faster and that will be a pleasure to use in filming.
Probably this article will be helpful to you mostly with videos with DSLRs.

http://photography-with-any-camera.blogspot.com/2013/03/video-recording-with-dslr-is-it-worth-it.html

DSLR's can't film. They're digital.

Video? DSLR's were never meant to do video and more importantly there's no dSLR specifically made to do video. Go ahead, ask around. The D600 is for shooting pictures. In the same way there's no dSLR lens specifically made for video. Wherever you are getting all of these should be burned at the stakes.

Yes, you can use the 50mm for shooting video if it satisfies the requirements for that scene. There's no difference between pictures and video on a dSLR. You use the same lenses.

Currently all of Nikon's 50mm lenses are FX, so it should work fine on the D600. The fact that it was previously mounted on a DX camera would not matter. And of course the D600 has the video functions it has.
If you happen to have other lenses for your D200 that happen to be DX lenses, then:
For stills the D600 has options to either use a cropped portion of the sensor to give you a DX photo, or to use the full frame revealing the dark corners that the DX lens does not cover.
For video, the DX lens on the FX body can work, but according to some, not well.
http://www.dpreview.com/...t/50068261