The D600 is very much like the D7000 in terms of body size, controls, and navigation. The AF system is a bit better than the D300, but the number of AF points can be a problem if you use the Shutter to focus.
I've used the D600 and D7000 enough to be very comfortable in giving up my D300. My primary camera remains the D800E, but I don't see any reason to avoid the D600.
One thing to keep in mind is that the corners tend to be soft on FX in comparison to the same lenses on DX. Version 1 of the 70-200 is a question for me and I expect I'll ultimately upgrade. The 16-35 f/4 is similarly a soft in the corners on FX cameras.
I do have a 16-85 and 70-300 for a light weight kit. The 70-300 works well as an FX lens and the DX crop on the 16-85 makes an easy transition.
I have not really pushed the ISO levels, but you pick up at least a stop or more compared to the D300. ISO 1600 to ISO 3200 was great for star photography recently and noise was almost non-existant.