With regard to the DV and sexual assault, were they shown to be false and is that documented?
In many states, not sure what state you're in, due to your not mentioning it, a person who has been found guilty of DV is not permitted joint custody. The laws vary from state to state so you'll need to research the laws in yours.
Many states have their code online that you can find using Google.
Somewhere in the Best Interest of the Child Statute or Custody Guidelines, it's going to mention this.
I would hate to give you hope on anything, since walking into a family law court is a crap shoot, at best. I really wonder if there is a need for all the court room hearings, since it seems that judge makes the wrong choice anyway. IMHO, it's a bunch of work and money spent all for naught... The only people who get rich in a custody/divorce are the lawyers. Get ready to sell a kidney on E-Bay...
However, there are many, many articles related to what you're about to go thru in the articles section of this site. Read them all... Read them twice and take notes. If you follow the advice they give, you increase your odds tremendously with regard to gaining custody of your kids.