I had to have TMO move my weapons to/from Alaska. You can't drive pistols or EBRs through Canada. I had no issues with TMO. Lock up your actions, slap on a trigger lock, then lock the case. Usually you only have to worry about TMO breaking stuff, not losing it!
For single AF officers not at UPT, you'll usually live off base. When I was in Alaska, I lived off base and it was cool.
If you're single and going to UPT, you'll most likely be stuck on base in dorms, unless you're lucky like me. I went to Vance and they were/are in the process of renovating their dorms, so I was fortunate and was able to get an apartment due to non availability. You'll probably have a couple of married dudes in your class that you'll be able to leave your firearms with off base.
I wouldn't bring your weapons to ASBC, leave 'em at home or wherever.
I don't think I've ever heard of ammo not being allowed in billeting/dorm rooms. If anything check the Air Force Inns website or call them up. Ask for a list of prohibited crap. If it isn't on their list, you're good to go. (But whatever you do, don't ask them about ammo because the person working the desk will probably assume you aren't allowed to have ammo in there.)
Also, remember, if you have a disassembled firearm, its just "parts". Nothing illegal about having firearm parts on base.
I checked in some pistols and a complete AR and an AR lower to the Armory at Tinker when I PCSed here (stayed at billeting for about a week). The cop at the Armory told me the lower wasn't a complete weapon so he said just keep it (saves him paperwork and me some worry).
Also, if you do leave them at the armory, I'm pretty sure you can put your own trigger lock/action lock on the weapon so they don't fuck with it.