Regarding your offsite car rental, while you have all of the extra fees associated with an airport rental (and they are high at BNA), you can often find rates that are low enough to offset the additional fees. Keep an eye on rates at on-site agencies especially in the days leading up to your trip, good chance you can get a cheaper rate and have the convenience of picking up and dropping off onsite.
Regarding hotels - I've stayed at the Radisson, Country Inn & Suites, Sheraton, and Holiday Inn near the airport on priceline stays. Depending on size of your party, this is likely the best way to go (if you only need to accommodate two people in a room), I've always been able to find a room under $100 before taxes through either name your own price or their express deal offerings.