Does paying for childcare count as a maintaining a household? I am unmarried with a child, but live with the child's father. I am claiming our child as a dependent.

No, paying for childcare does not count as maintaining a home.  Maintaining a home would be paying food, lodging, and general living expenses. 

No, only one person can claim head of household if you only have one child. That person needs to pay for more than half of the living expenses for a qualifying dependent in order to claim head of household.