If you both the house, he has a right to be there.
So it's important to be clear and straightforward and consistent about what you want from the start.
Bearing in mind that
He has cheated - male or female, it's the same thing.
He has lied and deceived you from the very start about who and what he is.
He is homosexual = you are not what he wants.
He has enabled the other man to knowingly do the same thing to his own wife. Two families devastated.
He is potentially acting in ways that could impact his own and your health - maybe an STI for you would be wise.
You're in shock now.
Short term - what are you going to do this week? He says he 'can't' go to parents (maybe tell him to suck that up as it's not your problem) and friend's place is not available, so the alternative is he finds a budget hotel/B&B for now. This will eat into family finances.
Christmas day - the children need to be considered, and they will want him.
Your feelings matter. Cooking and hosting him will probably make you feel awful and it will either be noticed by the children or you'll push yourself to unfair limits trying to hide it.
He may try to take advantage of the Christmas spirit and try love bombing you with presents, apologies, promises and tears. Remember what he stands to lose - that's what's driving that. If you cave - remember what he did and will do again because none of that will change.
I think the healthiest thing will be to start having the conversation about his long term plans asap.
Looking for and renting a small place close by so he can be present for the children - but not in your face.
What's happening with the other man? Is there the prospect of them setting up home?
I'm so sorry. All the best to you.