Legal Question in Family Law in California

It would be a good idea to file and serve both the paternity petition and any ex parte orders to keep the children in Calfornia.

Consulting with an attorney would be helpful. Many attorney will do an initial consultation for little or no cost.

Yes. You have no right to restrain a person from leaving the state with their child when you have no legally recognized connection to the child. You can, however, get a temporary order that the child not leave e state or only leave under certain conditions and assurances, until the paternity case is decided. This is not, however, something to try to do without a lawyer.