Adultery & custody are not related. Why you are divorcing has nothing to do with his ability to parent as far as the court is concerned. He has the right to request up to 50/50 shared parenting (regardless if you breastfeed or not - there are pumps) if he so chooses. It will be up to you to prove he isn't fit - and being a cheater doesn't make him unfit. Kansas is, for the most part - a no fault state - his adultery isn't even really going to affect your divorce. If you can prove he spent money on her you can demand reimbursement of your share of the squandered marital assets - but that's about it. You will need to request a morality clause in your parenting plan to ensure he doesn't bring the child around his girlfriend if that is an issue for you. (I'm assuming it will be)
Regardless if the house would be difficult to sell, if you can't refinance it into your own name and afford it, and he doesn't want to keep it and buy YOU out of your share (if there is any equity) it will have to be put on the market. If you kick him out, he can stop paying anything until a judge forces him to, but any payment he is ordered to make, other than child support, is going to be temporary.
My suggestion to you, you need to figure out a way to become finacially independent before you file. Come up with a solid exit plan. Once you file, even if he is ordered to pay things, he may not & it could take months to get any sort of money from him. It all depends on his reaction to it.
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