There is no requirement and no provision to register ordinary rifles, shotguns or handguns in Ohio. When buying a gun from a dealer, the dealer makes a record of the purchase on a Form 4473, which they keep at the shop.
Impossible to answer without more details.
No, Ohio doesn't require gun registration.
You don't need the Form 4473 unless you are purchasing it from a FFL holder.
No
There is no requirement or provision to register a shotgun in PA
It appears that Ohio does not have gun registration, so the answer is, you don't. You should verify that with a reputable local gunshop.
Contact your local police.
Gun registration is regulated by state. There is no federal gun registry, yet. You have to check with your state's gun laws.
As long as the serial number is on the gun and can be ran, you should be able to register it. If the number is bad, the police may confiscate it.
No
You don't. Like MOST states, Ohio does not require registration of firearms. http://crime.about.com/od/gunlawsbystate/a/gunlaws_oh.htm