You are in the worst possible position a single mom could be in. The first thing you need to do is to go after the father of your children. I know that will add another thing to your full plate. Over the last 35 years I have helped a lot of moms collect child support. Sometimes the fathers had to go to jail, Most legal aid offices will help for free.
You need to run not walk to an attorney. These type of allegations sometimes get a life of there own and are blown out of proportion. This is especially true when you have special needs children. DCF will jump all over these alligations.
If your case was dismissed with prejudice, then the lender cannot go after you for anything. When the statute of limitations runs out, (I think it 5 years) you can sue to quite title and be free of all encumbrances on your home.
It is hard to prove a negative. That he never hit her. She can claim what ever she wants. It does not make a difference if she yells it from a tall building. What counts, is if she tries to use the court system. Then all the proof you can get is important. You must make copies of all Facebook and the like.
That is a very hard question to answer with the information given in your question. But as a general rule you cannot modify alimony for the reasons stated in your question. Usually there needs to be a substantial change of circumstances to get a modification. You may be able to take her back to court for a violation of the divorce decree.