Rooting your phone only grants the user system privileges to the device. If you want to unlock your handset, you have to flash an unlocked carrier-free custom ROM in order to do that. Don't even get me started with the VZW GSIII's locked bootloader.....
No. All root does is give you access to every file on the phone, and the ability to change them for better or worse. I don't think the Verizon GS3 has GSM bands in it? I could be wrong, but even so, you still have the sim card locked to Verizon. Even if the Verizon GS3 has the radios to work on GSM, you have to get Verizon to unlock the sim first. You would be better off just selling the one you have and buying a GSM phone that will work where your moving to be honest with you.