I have told him I have no issue giving him 50% parental custody but I believe as her mother I should have sole physical custody, and he should be awarded visitation rights. However I believe he only wants physical custody to try and avoid paying child support. I have allowed him to see the baby whenever he wants, and I take her over to his parents house every Sunday so she can spend time with her Grandparents. Also I am perfectly fit mother very dedicated to my child. I have also had to stop him from trying to play with her by tossing her in the air at 3 weeks old, and giving her a very loud raspberry on her belly at 2 weeks, I don't feel he is mature enough to take her over night while she is still so delicate. What I want to know is what are my rights as her mom, and what are his.
Child Custody Lawyer
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.
Divorce / Separation Lawyer
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.