Nikon D600 now is fully supported in Adobe Lightroom 4.3 and Camera Raw 7.3 which are now available for download in their final form. In addition to the Beta camera profile left over from LR version 4.2, (which remains in the list), the Adobe Standard, Camera Landscape, Camera Neutral, Camera Portrait, Camera Standard and Camera Vivid are now in the calibration menu.
Be aware that the Camera Raw download is quick and painless, but the Lightroom 4.3 download is massive, taking about 4 hours and 45 minutes over DSL. Installation adds about another 15 minutes, so set aside some serious time for this. It completed without any problems on my 64-bit Windows 7 computer.
And I checked it for Tethered shooting. NO, the camera is not detected (although it finds my D7000 just fine)!