All US version seem to be the same hardware (sans the Sprint version not having a removable sim).
To me, this means that once you either have VZW unlock the device, or you find other means (there are quite a few methods being discussed on the interwebs) you should be able to use it on any US carrier. The US version supports HSPA+42, whereas the Euro version only supports 21mbps.
I am not sure...I've read that each carrier has an entirely different part number. And I don't mean like 747V, 747T, I mean like 747, 517, etc. Also at the XDA forums each carrier branded device has it's own section, leading me to believe that they are not created equal.
From my understanding. The Snapdragon S4 used in North American markets contain the same modem capabilities and functionality can be limited only by software. I also believe that Verizon had already said that they will provide "global capabilities" to their GS3 at a later time via Software update. As far as I know Qualcomm did not make separate hardware modems in their S4 for at least the U.S. based carriers.