I have done a newborn domestic adoption, so it's not quite the same thing, but I do have info on some of things you asked.
Our dd was born in IN. She qualified for medicaid at birth for her first 30 days. Once we returned home (FL) her medicaid was not accepted even though she was only two weeks old. I would really look into seeing if OH will accept it or if you can get her either on your insurance or see if she is eligible for OH medicaid.
For our domestic adoption, we had to be fingerprinted and send the fingerprints to the FBI. They were not sent to a FL office of the FBI, but rather one in DC (I think). It took 6 weeks to get clearance, not 6 months, so hopefully that will be the same for you!!!
I hope I was able to help a little!!
Oh, CONGRATS on your match!!!! You must be so very excited
Casey