Yes, normally an insurance company can subrogate, seek to recover from the at-fault party the monies paid out for the damages that person caused, if that person was at fault and uninsured at the time of an accident so that the "insured" person's insurance company had to pay out.

When an insurer is required to pay out money to an insured claimant, it is usually allowed to sue in the claimant's name against any person responsible for the loss. So in the scenario you described it would be the at fault uninsured person that could be sued by the insured's insurance company.

As for your second question the Ohio Bureau of Motor Vehicles notes that if in an accident the at fault party is uninsured and caused damages, they may have their license suspended until they pay (or until they make arrangements to pay) for the other party's damages. This likely holds true for it the insurance company requests payment and you refuse to pay or reimburse them for the expense of the repairs and damages.

For more information on this topic try contacting the OH BMV and/or Ohio Department of Insurance.