There is a lot to answering this question. The short answer is that if he has not filled out an affidavit of parentage then his rights maybe limited. You will need to get an attorney involved and file a petition to establish custody child support and parenting time and possibly even to establish him as the father which is called filiation. This will be a bit of a process hand while it can be done without an attorney you will get the best results in the end by having someone who knows what you are entitled to.
This answer is intended to help you understand your legal options but I am not able to give legal advice to you unless I had all of the specific facts about your situation and we had an attorney/client relationship. An answer like this is no substitute for meeting face to face with an attorney and getting actual legal advice, which is what I recommend.
Unless you are married until he is established as the father he has no rights in court. Once in court if the two of you do not agree on visitation it will be up to a judge to decide. One of you will probably be designated as the parent with physical custody. As for preventing overnights, that will be very difficult as you would have to show that there was some danger to the child not that he is immature. See a local attorney to get guidance on how to proceed.
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This advice is not meant to create an attorney-client relationship and is a general anwer to the question posed.