They're leaving a lot of things unsaid in this chapter and it's confusing a lot of people.
1. The fake ID doesn't have to be stolen, it could be someone completely made up through record falsification (too conspicuous for A rank) or the pseudonym of an actual 3-star magician who works with them (not strong enough for A rank). Technically, Frey just used the latter method to create the high-ranked fake ID he needs. It's possible they covered up the death of a real C rank adventurer and stole his ID, but that's already risky for various reasons, impersonating a living adventurer would just be stupid.
2. He's not trying to go undercover and infiltrate anything, he just doesn't want people to know Frey Blake is around. Nobody in that territory is going to hear about a new 5-star A rank and think "oh I guess Frey's back." He's in a hurry and is leaving shortly anyway, he'll probably ask them to keep it under wraps until then.
3. The wall is just a makeshift training room wall that was magically strengthened, it's not hiding anything.