timing on filing for probate and possible medical malpractice suit

A competent personal injury lawyer cannot adequately evaluate a case based upon information posted on a forum such as this. My best advice for you would be to contact a competent and experienced personal injury lawyer in your area. Only after gathering all of the information about your dad and the circumstances surrounding his death would a lawyer be able to adequately and fairly evaluate the case. And regardless of how intelligent you are, a layperson simply does not have the necessary knowledge to handle a medical malpractice case on their own. Medical malpractice is a very specialized area of practice, and there are many lawyers who are not even competent to handle them.

As to the statute of limitations, you are incorrect about it being two years. Under Ohio law, you only have one year from the date the cause of action accrued to file a lawsuit for medical malpractice. See R.C. 2305.113. However, because your father died, there are two statutes of limitations applicable.

First, for your father’s conscious pain and suffering from the date of his injury until the date of death, you have one year from the date the cause of action accrued to file a lawsuit.

Second, for the wrongful death claim, which is for the injuries and damages suffered by the surviving family due to the decedent’s untimely death, you have two years from the date of death to file a lawsuit against all responsible parties.
Also, you must have an executor/administrator of the estate appointed by the probate court before you can even file the lawsuit.
Hope this helps. You should contact an attorney immediately, as time can quickly run out, and it takes a considerable amount of time to properly investigate a claim and get a case ready to file. This includes the need to have the case reviewed by an expert(s) physician to obtain the necessary affidavit of merit required in all medical malpractice cases.
Best of luck, and very sorry for your loss.