When Christine takes a closer look at the photos, she sees that some of them look fake. Some are probably staged.
The real Ben likely put together a photo album for Christine to remind her who she was each day. These would logically have been among her possessions when she was moved to the institution.
Ben/Mike takes her to the hotel because he is tired of the pretense and just wants to be Mike. He takes her back to the hotel because he wants to "start over" and rewrite their history.
I think the timeline is hazy on purpose. Christine doesn't have a clear handle on the timeline, so you don't either while looking at the story from her point of view.
One of the points of this story that makes it a bit more compelling is that no character is completely virtuous. They all have some misdeeds from the past that they would like to erase or forget. Everybody lies or omits important facts at one point or another.
No one notices because they are distancing themselves from Christine. Fake Ben tells Christine at the hotel that he was able to take custody of her because all the other people who were close to her had been driven away.
Dr. Nasch gets the photo of Claire from Christine and presumably tracks her down through hospital records. There is also an implication that Claire has encountered the doctor before.
It is plausible that no one thought anything of Christine living in the house. Many people don't know their neighbors really at all. In real life, an abductor held two girls captive in his home for years in a densely populated neighborhood. Police had even been called to the house several times and the girls were never discovered until one of them escaped.
Adam's drawing appears in one of Christine's distorted recollections. I'm pretty sure Adam never sat at a desk outside the hotel room where Christine was beaten, either. She also thinks she remembers the doctor as her attacker her in the same "memory" sequence.
Are you talking about a case in Georgia because that was hella scary I hate watching the news especially cases like that that seem like a movie but it's reality. Makes me afraid of my own neighbors. There was a recent report about a lady found on a property locked inside some kind of contraction and was found due to her screams. She told police there were several bodies buried on the property. I don't remember if it was in Georgia though. But yeah this movie is messed up. Nobody thought to find out where the heck she was and why the doctor didn't know he wasn't her real husband???? Loop hole??? Torture porn??? Troll???
The real husband and friend Claire never noticed that Christine was checked out of the institution by "her husband"?
This is the biggest issue of all in my opinion, especially with the dialogues hammering in how much real Ben and Adam loved their wife/mummy. They loved her so much that in 4 years they never ever paid her a freaking visit at the hospital or inquire about her on the phone! Inglorious bastards!