<---jealous. (The locked bootloader in the AT&T version makes life difficult.)
Yes, it will work, once it's unlocked.
No, you can't get an OTA update unless it's currently on an active TMobile plan.
You can, however, manually update it with a captured update file (see
T-Mobile Galaxy Note 3 Android Development - XDA Forums) any time there's an update. (I wouldn't put 5.0-5.0.2 in, though - they didn't fix the memory leak until 5.1.)
And you can put any TMobile Note 3 Cyanogen ROM into it any time you want. (I'd also put TWRP in it.)