I have a Nikon D7000 and am going to buy a D600. I own a brand new, unused lens, an AF-S DX NIKKOR 10-24mm f/3.5-4.5G ED, as well as two FX lenses, an AF-S NIKKOR 70-200mm f/2.8G ED VR II and AF Nikkor 50mm f/1.4D. Should I sell the AF-S DX NIKKOR 10-24mm f/3.5-4.5G ED? I plan on keeping the D7000 as a back up camera.
I would sell the 10-24 and get a good FX standard zoom or wide angle since I would bet you will be shooting the D600 in wide angle situations more than the D7000. But I would see what I shoot mostly and then decide what you need to add to your kit before you sell and buy more lens.
I was in the same situation when I bought my d600 and decided to keep the d7000 as back up. IMO, I think it's better to have a lens that can be used for both camera bodies. I didn't have the 10-24mm lens but I had the 18-105mm lens and I had to sell it to buy the tokina 16-28mm f2.8. If I want a little more reach for widelife, all I have to do is put the 80-200mm to the d7000 and still can use the 16-28mm on the d600 for wide angle.
One argument for keeping the lens is that it is smaller, and therefore more portable, to use it on your D7000 than using an equally wide lens on your D600. You might want to use the D7000 with that lens as your main wide angle camera.
But, here's what I'd do. Keep the lens for a while and see if you use it. If not, then sell it. That would be preferable to selling it now and then wishing later you'd kept it.
I view the 10-24 as a bit of a specialty lens but it does cover a useful focal length. It's roughly the equivalent of the Nikon 16-35 which is one of my highly used lenses. I don't see anything else in your current kit to cover the ultrawide end and I think you need to cover most of that focal length.
I'd probably keep it. There is nothing wrong with using it in DX mode on the D600. It is part of a light kit. Until you have an alternative, I'd hang onto it.
In terms of alternatives, I have and like the 16-35. It's a relatively light lens and wide enough to be useful with your D7000. If you pick up a 16-35, I would get rid of the 10-24. I think you'll find that when you want to go wide, you'll choose the D600.
The 24-70 is a very good zoom lens - my most used. But it is relatively heavy and not really wide enough for DX. With the 24-70, you'll need to keep your 10-24.