I know that the D800 has an option to tune AF. Does the D600 have this option, and does it work as well as it works on the D800? Can I fix a back-focus issue with my lens on a D600 without sending it to a Nikon service center?
The Nikon D600 has AF fine-tuning and it works exactly the same way as on the D800. It will adjust focus front or back a number of tiny steps. The step size is unspecified so you have to do it by just trying. If your lens is severely off, it is possible that no number will be suitable and you will have to have it calibrated for your camera by Nikon.
Note AF fine-tuning is a user-driven process which lets you offset focus by a uniform amount. This means that you may not get perfect results if your lenses shifts focus at different distances or has strong field-curvature. Meaning that while focus may improve at one point, it may get worse at another. Some very recent Sigma lenses allow you to set the AF fine-tuning at various distances and interpolates between them which deals with the former problem but not the latter.