This is messed up. Based on your story I am certain that this dude isn't the most socially well adjusted of individuals. Not dangerous or anything, but, you were his first girlfriend and engaged to him? Failed engagement to boot. And he doesn't have any other friends now because he got a mortgage? but he's dating another girl? What?
Frankly, a serious engagement is not exited to a casual friendship (maturity issues?) as for the dogs, you gave them up on the break-up, let them go. The fact that he is not talking about his current girlfriend suggests there is an issue there that he does not want to talk about (did he break up with her, is he about to?)
Going out drinking with him and listening to "you look good, would it work out if we got together now?" is kind of telling as well.
A lot is not being told by him to you or you to us, it makes it difficult to give any sort of advice worth while. If you want to break up with him, do it, drop the constant IMing, following his social life on facebook, going to the pub and occasional phone call. If you don't, then get back together. If you want him to be friends and not lover, you shouldn't have slept with him in the first place, that's not an option, you can't go back. Especially if he has no experience with those sorts of emotions.
As it stands, I'd lay money he and said girl never went out at all, it sounds like he wanted to date her, I'd be willing to bet what's going on is that he asked her out and she shot him down and people are misreading the courtship as an actual relationship.
Male perspective, the problem sounds like it is totally coming from you, whatever feelings your harboring, you don't sound like you're being completely honest with yourself.
Besides, asking questions like this, you have already made up your mind about what your going to do and just want one of us to say that it is the right decision. We've all been there.