Re: Root question.
There are really two ways to go:
TWRP
CWM
Depending on the ROM and Kernel you choose to use, each developer builds based on what recovery method they find best. Basically when you load a ROM, you are loading an entirely new operating system for your Note 2. Most ROMs come pre-rooted and free of bloatware. Make sure you do your homework and understand what setup you are running to ensure you load tools that will work. You will have to involve your computer to load the CWM recovery method which is flashed with Odin, but you can load the TWRP via GooManager directly from your Note 2.
I am running TWRP, CleanROM, and the Note2Core kernel, phone is running really good and I am very happy!! My setup is entirely done from the phone, no PC needed.