can he get custody of our daughter with domestic violence on his record?

I am leaving my fiance and trying to get my ducks in a row to seek primary custody.

As of now, I work full time and he watches her while I am at work. I cant afford daycare until my mother moves in with me in October. He has 2 incidents of domestic violence (1 with an ex and 1 with me).

1) Will I look unfit for having him watch her while I work?

2) Will the domestic violence cases hinder him being able to come after me and seek full custody?

3) Should I seek child support even if I can provide financially for my daughter?

I am a new mom and didnt think i would have to do this, but i need to get my daughter and i out of this situation.

Please help me. Any advice is appreciated!!