You can flash the zip version from TWRP or the tar.md5 version using Odin.
The bootloader on the North American s3 is not locked.
The easiest way to root in my opinion is to flash a custom recovery and flash supersu.
Installing a custom recovery will trip the warranty bit.
Install the Samsung Info app from the Play store and run it to get detailed information.
You can also use a terminal app and run getprop ro.bootloader on a command line to get the bootloader information. The bootloader information is the most imp...
I would install a custom recovery and flash a custom rom. There are custom ROMs based on stock Samsung ROMs.
Thanks for updating the post:)
It's important to let other know how different issues were resolved.