I recommend Philz Touch because it is easy to flash it using Odin. I used to have TWRP and had problems flashing ROMs. Once I switched to Philz, I didn't have anymore problems.
The best modem in your case would be the mjb version. Even though the Canadian and US versions of the s3 are identical hardware wise, the bootloaders and modems are not interchangeable.
So what exactly would you like me to flash though (links would be nice if you could)?
As I've said, I've already tried various ROMs/Modems etc, the problem is deeper than that.
Edit: Flashed MB2
All of the same issues still occur. :'(
I747UCUEMJB_2024954_REV04_user_low_ship.tar.md5.zi p (I guess this is what you were telling me to do all along huh?)