If CPS has been unable or unwilling to take any action, then you are right to have filed a custody petition. You also are correct, however, that a judge is unlikely to move three children into your one bedroom apartment. This is the type of situation where a judge could look for other family members to care for the children, or even foster care, if you are unable to relocate and the father is found to be abusing the children.
Charges that are nolle prossed would not be considered by the court. If you know the facts surrounding the incidents, then there can be testimony about what happened. This could be relevant, depending on the facts.