True, never married couples, whoever the child is with has custody and its usually the mother.
Because there is no custody agreement, you can visit/move wherever you want, as often as you want.
If he ever were to call the police or something crazy saying you are keeping him from his child (not because he really wants to see the child, just to be spiteful) , the police would ask him for the court documents showing you are violating a court order, if there is none the police cant enforce anything.
If you need a passport there is a section where you can file saying the childs father has not seen the child in so long, you have to give last known whereabouts etc and it takes a lot longer but you can get one.
However name changes are complicated, if the father is ok changing it then he can sign the papers, however if he is not you will need to get a lawyer.
Been through the same things with my ex, now im married and have a family with someone in the military, we've moved around a lot and had no problems with anything expect the name change.
You can find an attorney online:
You can contact your State Bar Association and ask for referrals.
If there are local organizations that provide free immigration counseling or services for a discounted price consult with them first, and if necessary ask for referrals.
Another thing you can do is to go to immigration court and approach an attorney and ask if he/she might be able to help you.