can a registered sex offender in ohio win custody of his child?

It depends what the crime was, when it was committed, and the facts surrounding it. I do not think there's a provision in Ohio law which absolutely prohibits sex offenders from getting custody of their children after they have served their sentence, but if the sex offender was convicted of raping a child or something like that, I seriously doubt they'd ever get custody. It is a sad fact that today many people are labeled sex offenders who's crimes have nothing to do with deviant behavior. The label's been overused and abused by politicians and so called "victim's rights" advocates that it has become a catch all category for such a large group of offenses that aren't even related. However every time someone get's that label it's still another ruined life.