If you honestly believe your children have damaged your 5 year old home to the extent that it cannot be sold, you need to rethink your parenting style.
Don't listen to your soon-to-be ex. Have a realtor out to the home when your spouse is not there. He/She can give you an idea of the saleability of your home.
It is in the CHILDREN'S best interest to have court orders for custody and visitation. Ideally, each of you would have an attorney representing your interests so even if he does not want to hire one to represent him, you should hire one to represent you. It will be less expensive in the long run to do it right now rather than try to fix it later. And, don't stay on the mortgage and title with him in possession of the property, and definitely do NOT give him full ownership of the house but remain on the mortgage. If he cannot refinance it in his name alone, it will have to be sold.