For your subsequent question, you can file a declaratory suit to declare the child's name with your surname.
If you are the only parent filing a petition (request) to change your child's name - whether the other parent agrees or not - you will get a court date and you will have to let the other parent know about your petition and about the court date for the petition. If your child's other parent does not agree to changing the child's name, he or she has the right to oppose your request. If parents cannot agree on a child’s name, the court will decide based upon the best interest of the child. If the judge approves your request, you will get a court order called a "decree" changing your child's name. The court process can take up to 3 months, though in busier courts it may take longer.