The fact that he is not on the birth certificate and that you were never married is irrelevant to custody. There are two types of custody: residential and legal. While you have residential custody in fact (now you would be referred to as the primary parent) you have joint legal custody. This may be an issue for traveling out of the country, getting a passport or making major medical decisions, especially if you lose track of him in the future. I agree you should file for sole legal custody to avoid this. It doesn't terminate his parental rights, however. This means he can always file for parenting time in the future if you don't allow it.
973-984-0800. Please be advised my answers to questions does not constitute legal advise and you should not rely on it, due to the fact that we have never met, I have not been aprised of the facts in you case nor have I reviewed any documents.
No. Legal custody is established by the court. It actually doesn't have much significance. I believe that what you are actually trying to refer to is residential custody. If the father is not trying to be a part of the child's life, it may be a good idea to go ahead and petition the court for residential custody now. The court will likely on the same day of your custody hearing, order that he pay child support, if the court finds that he is the father.
You can petition yourself, but it always a good idea to consult a family law attorney before proceeding with such a case. I wish you the best of luck, and please feel free to call my office should you wish to discuss this matter further. (732) 609 3915.