i personally never found my AF in live view to work great in low light situations with any of my nikon bodies.
when i went outside, it worked ok, but whenever i shoot movies, i always manual focus, and i never really use live view to shoot photos unless i am doing macro type of photography, and even then its on single servo AF mode and i select the focus point.
but yes in live view mode in low light situations, my D600 would hunt and eventually would find the subject, but so would the D300 i used and my old D3100.
I heard Nikon is coming out with a firmware update for Live view mode for the D600..not sure when though.
for the continuous issue, no i havent experienced that at all.