I have now downloaded DNGConverter 7.4 as I believe it is the latest version that is compatible with my Mac 10.8.5 OS. But, when I tried to open it, it immediately quits, and will not open.
What do I try next to solve this?
Lastly, since this DNG version is not the latest version anyway, will it actually open a Nikon Raw File from a new Nikon D600 DSLR?
DNG Converter 7.2 and later require Mac OS 10.6 or above, Please refer the system requirements section of the download pages.
The Nikon D600 has been supported since version 7.3 of the DNG Converter, so versions 7.4 and 8.1 also support it.
You should have no problem running any of those three versions of the DNG Converter on your Mountain Lion system. However, the fact that you are having problems with 7.4 leads me to suspect that soething went wrong with the installation or download, or that there is some incompatibility with something else on your specific system.
Try the current version, DNG Converter 8.1, as per the link provided by Ashutosh: