Can FX lenses go on a DX camera ?

To answer the question, yes.
A crop sensor camera will provide a 1.5x (1.6x for Canon) equivalent zoom. i.e. a 300mm lens used on a DX body will give the equivalent of 450mm. Also doing the reverse (using a DX lens on a full frame) may result to vignetting. The D600 AFAIK has a switch to enable compatibility with DX lenses.
Look at my lens listed below, all are FX lenses and work perfectly on my D7100
The image circle at the focal plane projected by a FX lens is larger than the DX sensor, it works but, the field of view would be similar to that of a longer lens (focal length x 1.5) without the magnification. Similar to using an 8x10 lens on a 5x7 field camera, though the crop factor would not be nearly as extreme.

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