Except Blackwall. There is a bit of messenger tag where she finds a discrepancy in his story, but she decides it's not a priority so she doesn't investigate.
Tsk tsk. Letting personal feelings interfere with the job. Bad Leliana! Cullen does say she has a blind spot for Wardens. Either from the Warden or Alistair, if the former was a douche nozzle.
There is also the note reused from World of Thedas up in the rookery of the real Blackwall that shows he wasn't in Fereldan during the Blight.
Well, Solas is a trickster god. It makes sense that he could avoid detection. Possibly she thought 'don't look a gift horse in the mouth' when he showed up knowing so much.
Pretty much. His updated postgame codex has her kind of admitting that. That said, knowing Solas he would come up with a lame excuse that everyone would believe. "I saw it in a dream" or "I grew up in a abandoned village talking to spirits so what?"
The game doesn't really do a very good job of showing her as a good spymaster in the main game, though she can have some successes on the war table.
I don't know how much she is involved with Skyhold security but the assassination attempts on Bull and Josephine within Skyhold were close enough that they managed to breach the same room/get thrown off of the ramparts was pretty bad on her part. She is mentioned in both cases so she does have involvement with security. She needs to get away from this whole spymaster business.