If he has all that stuff by then (a place for him to sleep, food for him, diapers, etc) and he can prove it, I'm betting they'll let him take him for an overnight. Providing he's not a druggy/abusive and whatnot.
11 weeks old and an overnight!? I guess I wouldn't know because my SO and I are together and we're on our 2nd baby.. That seems early especially when there isn't a crib for him at your BDs.
If BD gets everything together yes he can. A girl I went to high school has her son one week and BD has him the next. It has been like that since he was 6 weeks old and that is state ordered..
A father usually gets visitation by default as long as there is not proof of instability.
Elisa wasn't out of my sight for more than ten minutes until she was almost four months old. And I cried the first few times I left her with anyone else. Even my SO. I just missed her and was worrying non-stop. I would be freaking out if I had to let someone have visitation. I'm sorry, that must be hard.