European 22 day Adventure , Germany part 3

October 16 arrive from Rome 2pm,

Arrive where? Munich airport I hope. Do not try to drive from Rome to Hohenschwangau (note spelling, you will not find much info if you look it up under the wrong name)? https://www.viamichelin.com/ says this is 873 Km and takes 8h50 driving time (+ stops), Costs: Tolls €51.50,fuel €72.15.
Also, hiring a car in Rome and returning it in Munich will incur a large surcharge for international one-way.
I suggest you fly from Rome to Munich, and pick up your car there, and dump it for your last 2 nights in Munich (a car is useless in Munich).

Rothenburg ob der Tauber (full name, there are many Rothenburg's in Germany) will not take 2 days, but you can visit other places on the way there and back.

Flight- Rome to Munich 1125am-2pm

Distance too far to drive, and see what we want to see in 3 weeks. Only way is with flights Paris to Rome and Rome to Munich

Have Hohenschwangau in the morning and put your rental car to good use by having a round trip in the afternoon. The clock-wise loop to the wonderful church Wieskirche, via the impressive monastery in Ettal and the Linderhof (another of Ludwig's palaces) won't disappoint you.
Rothenburg to Munich can be made into a nice trip when you go via Würzburg (impressive palace) and Bamberg (nice historical centre and an impressive cathedral).