Hi mdoodles,
Read and caught up on your situation. Interesting for sure and I feel your stress.
As far as the "no contact" letter....why bother? If he could write it and it would magically cut off all ties with the OW then I would agree. But it doesn't. So tell him you would like him to let her know he's home, but that's up to him. Besides at this point the way she harassed you, she has to know. Let him dump her his own way.
I don't know that there is such a thing as complete transparency unless you can be with them 24-7 and hear every conversation.
I think at this point, if you want to trust him, you need to tell him. I would say "I want to trust you. So I'm going to. You are a good man who made a mistake, you owned up to it and I believe you will do the right thing by us. That means cutting things off with this poor girl. (That's right, feel sorry for her--she's pathetic) It also means I'm not going to bring her up anymore, she's done and out of my life. Thanks for working on this...We are worth it." And leave it at that.
Trust is consistency. Show him he can trust you and he'll want to show you the same thing. If you say you're going to open the store, go open it. If you say you are going to make dinner, make it. If you say you are going to be faithful to him forever, do it. But I think being consistent in the small things build to really good bigger things. He'll mirror you.
As far as the couch goes, I agree with some other posters, get him a blanket and kiss him good night. He's home after all.
Just be the best wife you can be, be the spouse you would want.