Question...

Once you add your SIM you should be able to go to the Mobile Networks and reset defaults

 

it is partly the sim, but every rom has a APN.xml in the /system/etc folder (linux speak) If your familiar with linux O/S you can find many interesting things on the phone.
So that APN.xml has a list of dozens maybe hundreds of possible apns your phone gets the network identity from the sim and picks the correct one (usually) sometimes though you need to manually edit the apn, but not to worry its not like your editing the xml file just a few taps on the menus of the phone and enter in the information into a form.

And yes leave the sim in you wont hurt anything.

 

Welcome to the AT&T Note on T-Mobile world. There are Some great ROMs that make the Note nice and speedy. I can't put this phone down! I'm getting TMobile LTE on it and am very happy.

 

Welcome to Tmo Prinny!

If for some reason you ever need to input your apn settings manually, you can find them here.