Wow! that was a lot of questions about the military. Let me start with Arthur's question about Personnel Files: They only deal with a person's military career. Anything prior to enlistment/commission isn't/won't be known.

Hellbringer: A few of us Military cats went around and around about where Saul transferred from. KC come on and basically said that He's never said Saul was "10th" Mtn... just Mtn. So, he's taking liberties with the Division. Creative license and all that jive.
In the part where Chinwe asks him to prove he's military. Chp 26 ep 3/3: Time hack @6:39 "Saul Tink. 11B based out of Bell." Is that Ft. Belvoir?? I dunno. I'm and USAF dude and not hip.

Blood&Ice Cream..
Not to sound like a dick, but being Jr. enlisted doesn't mean anything when it comes to MOS/AFSC's. If you're qualified, you're qualified. You have E2's doing target acquisitions downrange.
I'm not Army smart (heheh.. sorry.. interservice joke was bubbling up) but Mike and Saul can have different MOS and serve in the same Battalion. Come to think of it, Mike has never said what his MOS actually is.