Do you have to fly into the nashville airport to rent vehicle from airport?

I don't think so. Remember there are also Enterprise locations around town that may be cheaper. Airport rental facilities have to pay extra so you get charged more there.

It's in the parking

No you do did. Its like any other car rental company

You do not, but if you are using a debit card to reserve a car, you do need a flight itinerary. None if you are using a credit card.

Not sure. It's outside of security so you should be able to.