Help with Itinerary

One solution is to fly in and out of Munich, but I assume this is a ff miles situation. Car will not be much fun to have in Munich, more like a ball-and-chain.
I'd vote for itinerary 2. Fewer one night stands.

1 Night in Passau (or any other place of interest getting me close to Frankfurt)

Passau does barely bring you closer to Frankfurt than Salzburg. So your second itinerary seems less stressful. However, a car in Munich would be of no use and causes extra costs. Besides, you could do your itinerary entirely by train. If you want to have a car drop of it as soon as you arrive in Munich and rent another one for the rest of your trip. A good place near Frankfurt airport would be Mainz or a nearby town on the Rhine.

Since "Our must dos are Nuremburg, Munich, Bavaria (a lot of territory), and Salzburg", I'd suggest sticking to just four locations and skipping Rothenberg.. With only ten nights try to split them into two and three night stays depending on your interest in a particular area. The reason for the suggestion is that, in my opinion, either of your itineraries will consume a lot of time just getting from place to place. Perhaps a fast train from Frankfurt to Nuremberg and rent a car there for the rest of the trip.

Sam, we booked our flights already to Frankfurt. It may be a more expensive rental to drop the car in a different city than to pay for parking in Munich. I will have to check.

"Our must dos are Nuremburg, Munich, Bavaria and Salzburg."

Bavaria is not a separate entity from Nuremberg and Munich. Both cities lie within Bavaria, which stretches northwest almost to Frankfurt. See this map.

If by Bavaria you mean "the Bavarian Alps" then I'd suggest slicing off some miles by eliminating your night in and the detour to Füssen and instead visiting the Berchtesgaden area, either as a day trip from Salzburg (they are very close) or you could spend your 2-3 nights in Berchtesgaden and do a day trip into Salzburg (that would probably be my choice if you really want to enjoy the mountains.)

http://images.fotocommunity.de/bilder/bayern/bayerische-alpen/jennerblick-zu-watzmann-und-koenigssee-1a243f3e-5ae1-42a3-b7a8-d2366c66aa75.jpg

https://www.berchtesgadener-land.com/cdn/uploads/st-bartholomae-mit-boot-koenigssee-thcontentgalleryresponsive.jpg

http://www.berchtesgaden.de/en/cycling

http://www.berchtesgaden.de/en/f94e56f2-d6d6-fa6a-f252-f27d21c63147.html

https://www.salzbergwerk.de/en/salt-mine

http://www.kehlsteinhaus.de/en/

http://www.obersalzberg.de/obersalzberg-home.html?&L=1

https://www.tripadvisor.com/Attraction_Review-g187299-d3371293-Reviews-Wimbachklamm-Berchtesgaden_Upper_Bavaria_Bavaria.html

Transportation and itinerary: The car isn't much good in Munich, Nuremberg or Salzburg IMO. Driving on Day 1 after a transatlantic flight is dangerous (jet-lag, drowsiness, unfamiliar territory.) You COULD do this rather easily by train and avoid roads altogether as well as gas, parking expenses and driving responsibilities.

Train from FRA to Nuremberg for 2 nights (roughly €50-120 depending on your ticketing choices.)

Train to Salzburg or B'gaden for 3 nights (Bayern Ticket day pass, €31/2); day trip to other town (cheap.)

Train to Munich (Bayern Ticket day pass, €31/2) for 3 nights

Train to Rothenburg (Bayern Ticket day pass, €31/2) for 1 night.

Train to Frankfurt for 1 night (Weekend ticket or Day ticket for Germany - around €50/2)

Fast train from Frankfurt to Munich, do your sightseeing there (including train to Salzburg), then come back into Germany to rent car to take whatever leisurely driving route you wish back to Frankfurt and return car there. It doesn't cost additional to rent and return in two different cities in the same country (Germany) but it can be considerably more to rent in one country and return in another (thus returning to somewhere in Germany to rent the car rather than doing it in Salzburg).
You could sleep in Nuremberg and day trip to Rothenburg, Wurzburg, Bamberg or any number of other great towns. And the drive from there to Frankfurt isn't too bad either (you could stop at Bamberg or Wurzburg in route, even). I would limit the changes of hotels and embrace the day trips. Super easy to drive outside major cities.

Thanks for some great suggestions. May consider taking a train to Munich from Frankfurt and getting the car on our way out of Munich. We really want to have a car to explore the countryside and not be restricted by train schedules for the remainder of our trip.

Here is what I would do (we have done something similar just backwards)

Fly into Frankfurt
1 night in Rothenburg
1 night in Nuremberg
3 nights in Munich
2 nights in Salzburg
1 night in Fussen (I might delete this as it is kinda out of the way)
1 night in Stuttgart or Heidelberg (depending on how far you want to drive)
1 night near the Frankfurt Airport

I hate backtracking so that is why I am suggesting this order

Here is what I would do:
Day one, regional trains to Nürnberg (weekend, Schöneswochende, weekday, RMV ticket to Kahl, Bayern-Ticket to Nürnberg)
3 nights, 2 days in Nürnberg, see
  Nürnberg one day
  Rothenburg one day (use VGN Tagesticket Plus)
Regional train to Salzburg (Bayern ticket or Savings Fare)
3 nights, 2 days in Salzburg, see
  Salzburg one day
  Berchtesgaden one day (RVO Tagesticket)
Regional train to Munich (Bayern Ticket)
3 nights, 2½ days in Munich,see
  Munich, one day
  Fuessen, one day (Bayern-Ticket)
   Munich in the morning
Munich to Mainz ,afternoon, Savings /fare
One night in Mainz. There are three hotels right across the Bahnhofsplatz from the Mainz Hbf and the S-Bahn to FRA.
S-Bahn to FRA, fly out

There is really no need to stay near the airport before a flight the next day, when the city of Frankfurt is only 11 min. away by train. Trains run every 15 min. and start at 04:15 in the morning.
Get rid of the car early and spend the day sightseeing in one of the most historic cities in Germany. Hotels are cheaper in the city and you then have access to lots of restaurants and things to see and do.

Are you planning on driving to Rothenburg after a trans-Atlantic flight? That is a dangerous thing to do.