Overnights?
I know I am really looking to the future right now with this question.....my x hasn't seen his daughter since the hospital almost 3 months ago and I've gotten just 2 texts messages from him so I'm not even sure if he wants to see her. And no paternity has been set up yet. But I am applying for state assistance with daycare and the state will go after him for cs so in turn he would get visitation if he wanted it. Do the courts normally let a father take a child over night before their 1st birthday? I worry about her safety with someone who has no clue how to care for a baby.
On a side note can I just say you all are so nice and it feels so good to have somewhere to come and read and feel like their are other people doing this whole thing along side me. 
Re: Overnights?
Your X will need to file for visitation/custody before he can "get" anything. Just because CS is being filed, it doesn't automatically mean he has the right to visitation. That's something he needs to do on his own.
But to answer your other questions, generally the courts won't automatically grant overnight visitation to a child under the age of 1, especially if the other parent hasn't been involved.