Your father won't be in a position to kick you and your mom out of the house (at least not if your mom doesn't cave). If your father has a job, he's an idiot to think the state will not come after him for whatever public assistance you and your mom need to get if he won't pay support. (I presume your mother is seeking both spousal and child support.)
"The police were here earlier for a complaint that we were left home alone yesterday."
Well, given your age, I can't fathom why they'd even come by in the first place but I suspect the complaint wasn't just "left home alone." It wouldn't be unlawful to leave a 16 year old at home in charge of younger siblings even for days.
"... and she is now in trouble and might lose her job."
Well, I hope that isn't the case and that her employer isn't an idiot. She'd qualify for unemployment benefits if she gets fired for attending to an emergency like this.
"The police offer told my mom they are not going to do anything now but if he or his girlfriend call thje police again they can charged with filing a false report."
Well, that's good.
"The courts gave my Dad 50% temporary custody even though he is a day or 2 late in picking us up and everytime we go to his house he is out with his girlfriend and her kids are mean to us."
Your mom is free to bring this up in court.
"My mother seems reluctant to standup to my father and is afraid of him."
Then I would urge her to seek counseling about this and tell her that you understand she's afraid, but that her lack of courage is causing you damage.
"Am i allowed to talk to the judge during their next court date?"
There won't be a rule against it. You're free to pipe up and see what happens, but I'd let your mother know that you want to speak if the hearing is about child custody issues. If I were you, I'd pick an appropriate moment to say "Excuse me, your honor, but even if my mother doesn't know procedure or is afraid to rile up my father, my sisters and I would appreciate you letting us tell you what we think about this."