Yes, Ohio will have jurisdiction and the child's school district will generally be where the custodial parent resides assuming Dad remains out of state; even if Dad has joint legal he may have difficulty fighting your decision
I understand your concerns. Without wishing to sound harsh, this is how the court will see it:
You trusted him enough to sleep with; you're basically telling the court that he's fine Dad material. He doesn't need a car to see the child. You can make alternate arrangements with Dad, but if you can't come to an agreement the State will .generally order visitation based upon the guidelines.

