Liv, I have no idea what the laws are in your country -- I don't even know what country it is! But my SA states that child support can't be withheld for any reason. It can only be modified by the court. If your H thinks he doesn't have to pay if D is with him, he should understand child support is to pay for her portion of the household expenses -- which go on regardless of her location.
I think it's time for you to get tough with him. No, he may not modify the child support. If he argues, refer him to your L. No, do not agree to selling the house. If he argues, refer him to your L. No, he may not stay in your home. If he argues, refer him to your sister!
I think I'd like her.
Honey, your self-esteem took a beating. Now it's time to soar. You are an intelligent, capable person. Don't believe his tired story and his weird perspective. You know you are fabulous -- as do all of us here. It's time to live your fabulous life!