Regarding Auto Focus modes on the D600, there are three, Auto (A,) Continuous (C) and Single (S). If you pick Auto, the camera decides between C and S. If I had to guess, it picks single whenever it can.
Regarding the frames per second rate, let me offer you a different viewpoint. For any given level of cost, there is probably one data amount that can be run through the camera. If you have a bigger sensor (say 24 megs) you get 5.5 frames per second, or 132 megs processed. If you had a 13.2 meg sensor, then you get 10 frames per second.
So if you want a higher FPS, buy a camera with a smaller sensor, or step up and pay more money for better technology. (Regarding my numbers, ymmv, but I think the general idea is correct.) Think of the D800..36 megs at 4 fps = 144 megs per second. It might be faster than the D600 through put, or it might be the same (maybe it.s 5.6 fps for the D600) and look different dues to rounding.