I don't have either a D7000 or a D600, but I have made the transition from DX (D90 & D300) to FX (D800 & D4). I also have the 28, 55, 70-210, and 500mm lenses you list and have used several other MF Nikkors on my DSLRs. My observations and comments are:
I think the FX bodies are much easier to focus using manual focusing than the DX bodies when the bodies have the standard focusing screen.
For a D7000 I would recommend a Katz Eye replacement screen. I installed a Katz Eye screen in my D300 and it significantly improved manual focusing. The split image rangefinder and the micro-prism in the Katz Eye screen make manual focusing much more precise.
The D7000 will have approximately the same field of view as your 60D, the same lenses on a D600 will have a wider field of view.
I believe the first six lenses were all originally made for 35mm film so they should work fine on the D600. I have compared several of my MF Nikkor primes to my next to latest generation Nikon f/2.8 zooms. Most of them did as good or better than the zooms at the same focal length and aperture at both the center and corners.
You cannot easily adapt your Canon mount lens to a Nikon body, because Canon has a shorter lens mount to film/sensor distance than Nikon. This means that without some optics in the adapter, the lens cannot focus to infinity. Optics in a Canon to Nikon adapter would probably change the focal length of the lens.
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