I definitely would leave the same evening, as loosing one night in an airport hotel does not seem so charming. In FRA you should be out of the airport with your rental car in about 1:30 hours maximum and another 1:30 (186 kilometers, mostly no speedlimits on the highways) will take you to Rothenburg ob der Tauber (I assume you mean this and not one of the many other Rothenburg in Germany as it is in the direction of Munich). So after all you should be there at 10:20 approx.

Regarding rental cars you can consider one of the big companies like Hertz or Avis, whereas I personally would prefer Sixt in FRA. If you are looking for the best rate (and are a little bit capable of the german language) go for www.billiger-mietwagen.de - I have saved lots of money with them!

Have a nice trip
cma410