My question involves a child custody case from the State of: Florida

My boyfriend and I just broke up almost 2 months ago. The other day he mentioned something about custody of our 4 year old daughter. I told him that he’d have a fight on his hands and then that was it. Well now I’m wondering if he’s going to try and he won’t answer me if he’s going to try or not. My question is does he have a good chance. I've never really done anything wrong. I did get arrested but not arrested, they put me in handcuffs at the store for petty theft but let me go that same night at the store and I went home. I did everything I needed to do and the states attorneys’ office dropped it but other than that I have never done anything wrong but it’s just worrying me that he could still have a good chance because of that. His name is on the birth certificate. He does get her every Wednesday and all weekend every other weekend and then the other weekends he only gets her for a few hours each day and he has not paid me any child support. He said he was going to start giving me a money order every week for $50 so I can’t say he’s not paying me anything but he hasn't started yet and didn't know if he even will or not until I say something. He told me that he talked to a lawyer and that lawyer told him that he doesn't have any legal rights to the child even if his name is on the birth certificate except what rights I give him.