Support and visitation are two separate things. If he has not been legally deemed your son's father, he doesn't have an obligation to pay support. The other issues you mentioned will be up to the judge. Typically, unless you are a convicted pedophile or murderer, you will have some form of parenting time. Your best bet would be to suggest a schedule to the other side ahead of time.
Visitation and support are separate issues. However the issues you raise should certainly be brought before the court if the father goes to court to try to get court ordered visitation. If he has not been legally determined to be the father yet that would be the first thing he would have to do before he could pursue visitation.