Is there a chance that my abusive ex will gain rights to our kids if we never married and he isn't on the birth certificate?

There is always a chance that a court could award him custody. However, as a general rule, the court doesn't change custody unless there is a problem or you agree to change custody. As long as you don't neglect or abuse your children, the court should continue to give you primary physical custody. You can best protect yourself by hiring a lawyer to represent you. If you can't afford an attorney, you can contact some legal aid organizations to represent you for free. In any event, you can protect yourself in court by not agreeing to anything unless you want the agreement. As an example, since the children aren't in school, you may be asked to share custody and give father custody 50% of the time. Don't agree to this unless you want to. Don't let anyone pressure into an agreement you don't want. You may be sent to probation for mediation before seeing the judge. Don't let probation pressure you into shared custody unless you want this. You should also ask for child support. You probably should file a motion seeking child support.