Exactly. Booster didn't decide to come to either Bruce or Dick with the information. He had two chances to tell Bruce before it was revealed that Bruce already knew. He never admitted what he was doing without the issue being forced by the proof Bruce had. With Dick he was caught red handed twice. Both times he confessed because of it. I don't see Booster flat out telling Barbara because of the guilt, his mission (which was why he didn't leap to tell Dick/Bruce) and because Bruce is closer to her hence it would seem logical that he'd be the one to tell her. If not then the only way I could see it happening is if they were forced into a situation together, likely with her already knowing.
The "Bat God" is pretty much how Bruce acted during the Omac Project. Booster tells Bruce he's in the wrong. Bruce goes off to get his spy satellite back never thinking that he is the one in the wrong. That his reasoning was justified. He puts the mission in front of friends. Shutting people out, that kinda thing. In a way he was the Bat God by how he treated Booster knowing what he did/would do.
I think so too. But like the "is Ted alive" plot I don't want it right away. But the question to me would be how to bring it about if she wasn't informed. How would she find out? If she's in the cave now (so I've heard) then would Dick leave the file down there? Would she have a reason to look for Boosters' file anyway? So many questions.