A couple things come to mind that you could check once you get your replacement. In live view the D600 uses contrast detect autofocus (cdaf). I've found that at or near the minimum focus distance of the lens cdaf can struggle to get focus. It happens on both my OMD and my D600. And in low light situations the subject might not have enough contrast for the camera to get a lock on it even though the operator can see just fine.
For the continuous shutter I would check once it actually has focus on something. If it's not in focus then the camera might be refusing to take more than one shot because of that.