I am not a regular 'reviewer' on amazon, and there are plenty of great reviews regarding your camera and this camera. That said, I am a huge fan of this camera. The jump from the DX format (yours) to the full frame FX format (D600), is clear. For the price, you can't beat it. It is expensive, but as good as nearly any in the Nikon professional lineup.
I think the answer to your question is going to depend on several factors. First, are you willing to upgrade your lenses to FX lenses if you have a collection of DX lenses already? You could use DX lenses on this camera but then what's the point of getting a FX body if your not going to use the entire sensor? I think the biggest advantages you get from switching over from DX to FX is better image quality and lower noise with higher ISOs with the bigger sensor and on the wide-angle side you'll have lenses that get wider than what's available on DX (on the telephoto end you lose some reach). If you don't want to spend the extra money to switch out your lenses then maybe you'll want to consider upgrading to the higher end DX cameras like the D7100. Hope this was helpful.
I went from D300S to D600, which, in my opinion, is a good entry level FX-format DSLR with amateur controls (D300S controls are more flexible and accessable). I sold my DX lenses and am very pleased with new FX set, but very expensive. If your style of photography doesn't require FX, you may want to consider D7000, which is less expensive upgrade from your D3100.
There is no comparison between D3100 or even D5100 and the D600. I had the D5100 before and I upgraded to the D600. It is just marvelous... except that little dirt problem, I had to return it for cleaning up at Nikon, I hope I won't have to do it very often.
Absolutely! If want to get serious about your skills, it does make sense. Consider the size and weight though. The D600 is bigger and heavier, but it is also a much better performer with most of the Professional features. I upgraded from a fabulous D7000, and have not regretted it one bit. The star rating is at 4 stars, due to lots of ppl who apparently got units with sensor dust. The other complaint from ppl is related to the focusing points being mostly in the center of the frame. I have not even noticed it to be honest. Other than that, this camera ranks # 1 if you look at price, build quality, features, and of course image quality.