Don't let him live in the annex, you won't be able to heal and move on with him so close. It won't be healthy for you.
Petition him asap, citing unreasonable behaviour, more straightforward than adultery, and the important thing is to divorce the twunt. It's never an easy process, but don't make it more complicated than it needs to be You definitely have enough grounds for unreasonable behaviour.
I'd have papers served asap. Then I'd go low contact. Get everything tied up financially by the SHL
He may appear quite reasonable in the next few weeks, but he will change and get very hard nosed about financial stuff, so please don't agree to anything. Your solicitor and his can negotiate.
I do know what I'm talking about, my ex left a twenty year marriage and three teens. My SHL got me an amazing deal, and she handled everything. If I'd come to an arrangement with ex, which he wanted, we'd have been very badly off.
Best advice upthread: He's not your friend.
And he's not the man you thought he was. Don't trust him an inch.
And you must, must, must do your course.
Ultimately, this is shit. But him living in your annex has disaster written all over it. I bet anything that you'll have a course related weekend away, and he'll move her in there for a night or two. Don't let him shit on his own doorstep. He needs to be away, in another postcode!