1. The military is NOT an aviation employer per PRIA law. So airlines to do not do a PRIA record check with the military. The military is also not affected by the PRIA and is thus not required to keep or return records anyway.

2. The DD-214 is the gold standard as far as employer insight into your military background. There's good reason for this; military record keeping has always been haphazard and it could be hard or impossible to contact previous CO's or units which may no longer exist. The military also will not release most records to such a request. This is why DD-214's came about in the first place. Otherwise the process of obtaining information about one's military service would not be fair and equitable.

I have heard of some majors asking to see FITREPs.

I'm not sure there's a legal barrier to an employer calling your former CO/unit, although the CO/Unit may be limited as to what they can say. If I got such a call (especially while on active duty) I would talk to the JAG before disclosing anything good or bad, unless I was being called as a personal reference vice a CO/supervisor.

You can of course use previous CO's/supervisors as personal references, and I would highly recommend this. A good reference from a boss or three will go a long towards shedding light on your military service.