Frankfurt - Nuremberg - Rothenburg - Munich - (Dachau) - Hohenschwangau ("castles") - Munich - Salzburg - Salzburg countryside - Vienna - Prague
It appears that in 10 days you have 9 substantial train trips (10 with Dachau.) You'll have only 1 day, really, in Vienna, only 1.5 days in Prague, maybe a few jet-lagged hours in Nuremberg, only one day for Munich (and Dachau) sightseeing. And you are doing the check in, unpack, pack up, check-out routine 5 times. You need some streamlining to achieve some time in these places; otherwise, they are just goals you are checking off your list while you ride around on the train or in your car.
8/14: Munich in the morning for 4 nights. Day 1 is mostly a crapshoot (jet-lag? flight delays? ) You might make it out of FRA by 9:00. Just see what you are able to see in Munich. Or if you feel you are up to it, plan a stop on the way in the Romantic Road city of Würzburg, a town you must pass through to reach Munich anyway, and a worthwhile stop. Stow bags in a locker and visit the Residenz and adjacent Hofkirche, investigate the old town area. Total train time this day is about 3.5 - 4 hours (the same as with your plan to do Nuremberg and Rothenburg.)
8/15-17: Munich, day trip to Dachau, day trip to Salzburg (do this one on Sat or Sun so you can have a long day out using the Bayern Ticket daypass.) The day trip to Neuschwanstein would take 5 hours train + bus travel round trip (N'stein is not a castle, by the way, but a late 19th century palace, not worth considering IMO just for the $17, 30-minute tour.) You could visit the impressive Nymphenburg Palace (right in Munich) instead.
8/18: Nuremberg for 3 nights. The Kaiserburg castle complex, the old town, the Fraunkirche and Lorenzkirche, the Dürerhaus, the beer pubs, the Nazi Doc Center and Rally grounds, all in all a great place to visit and a good place for short, easy day trips.
8/19 & 20: Nuremberg, day trip to Rothenburg, or?? ( If you've been sold on the somewhat hokey Nightwatchman tour, you can see it and still get back to Nuremberg on the 10 pm train.) Daytrips to Bamberg (fabulous old-world city with UNESCO World Heritage recognition) and IPHOFEN (lovely old-world walled town with emphasis on wine and art - much more pleasant than overcrowded Rothenburg IMO) are also possibilities. Too bad there isn't more time.
8/21: Prague for 3 nights.
Vienna is a fine city but it requires more time to visit and to get there than you have.
There is no reason all of the above cannot be done by train (or in the case of Prague, maybe the new DB IC-bus, which is faster than the train.) The Frankfurt - Würzburg - Munich journey and the Nuremberg - Prague journey should be pre-purchased at the DB site as soon as possible for the best prices - your cost will rise if you wait. The other trips can be done by local daypass - just buy as you go (Bayern Ticket.)