Filing Taxes with a Dependent

I have a one and a half year old. His dad and I are together,  but not married so we file our taxes separately. I have insurance through my work, my son's dad has Covered CA, and my son was going to get covered CA through/with his dad, but he doesn't make enough and so my son had to get on Medi-Cal. 

My question is, if my son has insurance through his dad (Medi-Cal), can I still claim him- or because his insurance was based on his dad's income, would he have to claim our son? 

Also, it asks about how much custody I have- if we are together, would I just put 100%?