No. The state of Ohio did not invent the paintball gun.
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.
Depends on what state you are in. In MOST of the US, there IS no requirement to register a gun.
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.
Depends on the state's firearms laws.
Contact your local police.
You don't. Like MOST states, Ohio does not require registration of firearms. http://crime.about.com/od/gunlawsbystate/a/gunlaws_oh.htm
The state of Missourid does not require registration, so the answer is, you don't register it.
Depends on how the local and state laws are written.
Go to headcount.org or rockthevote.com to register, it will walk you through the process in about five minutes (literally!) for any state.