Can my daughter's father have his daughter stay with his girlfriend while he is in jail??

No. That doesn’t sound right. Go to family court and file a petition.

I have custody , what if dad don't sent kids back during visitation?

File a habeas petition in family court first thing if they are not returned.

How do I file a motion to file to see my child?

Go to family court and file a petition.

That is not enough of a reason for him to get custody. You should file for custody of your baby and ask for child support.

I need to know if I can sue the board of education and the school bus that my daughter was taking for neglect?

You need to act fast. Your time to sue has may have already passed.

Does a written agreement count as child support document in court?

No. He needs a court order for child support for you child or it will not count in court.

How how can a father put a child in foster care without him having physical sole custody of the child but the mother does

You need to get an attorney ASAP there are serious unanswered questions.

You are not legally obligated to pay his bills.

Can the non-custodial parent get in trouble with the courts for styling a child's hair?

This should not be much of an issue. The most the court can order is that you do not style the child’s hair and most likely the judge will say it...

She should have the child in New Jersey and you should take custody of the child at birth.