Wow. Well, it sounds like he has some major trust issues, to say the least.
I did a background check on my ABF before he moved in - can't be too careful, especially since I have kids. But I told him about it and explained why.
I am sometimes guilty of snooping on him - he has lots of friends who are girls and last year one of them was really, really going after him. Really - trying to get him to sneak out and meet her, the whole works. He handled it well - told her no, but it really threw me for a loop because he was texting her as "friends" for a long time before I found out about it AND he didn't tell me about her antics - found out through someone else. I felt like he had violated my trust by not telling me about the problem.
So sometimes I still take a peek at his call log and texts. I shouldn't. But, I do. I like to reassure myself that I have nothing to worry about. However, I would never go so far as to record him or do checks like that - that seems.... off... to say the least. It scares me a little bit for you - like maybe he is borderline abusive or obsessed.
If the bill was on the table then of course you can look at it, IMHO. My boyfriend and I do - we have a mutual understanding that if it is left out in plain view then it isn't private. If it goes in our drawers then it is private. We never talked about it - just something that we have an unspoken agreement about.
This is a tough one to handle. I know this is a delicate question but have you given him a reason to distrust you? If so, you might have a talk about it and come to some sort of understanding about transparency and being able to share your concerns. (And don't get me wrong - he is going to extremes even if you had done something truly heinous like had an affair!! Definite red flag there.)
If you have never given him a reason to distrust you, I would confront him. He needs help with trust issues and codependency. His monitoring behavior sounds like codependency and possibly anxiety issues. (I am speculating... I am not a mental health professional.)
But no one can live that way - it is controlling and he needs to stop, especially if you have never done anything to spark his distrust. He needs to either work out his issues and learn to let go or get out, in my opinion.