Munich to Fussen, Augsburg, Rothenburg, and Würzburg by train or rental car?

Can it physically be done? Yes. Does it give you enough time to see much of anything in those places. Nope, not really. Especially Fussen. Have you looked at the dbahn website for train times? It's a 3 hour trip from the Munich airport to Fussen. So it's unlikely that you would get there much before noon. If your goal is to tour the town, then that's fine. But if you want to visit the castles, that will eat up the whole afternoon. And you'll be doing this in a jet lagged state. You wouldn't get to Augsburg until after dinner. No time to visit there if you need to get to Rothenberg ( I'm assuming you want to see Rothenberg odT, not the other one) the next morning. You could spend a good part of the day in Rothenberg (along with all the other day trippers), but you'll miss the best part of a stay there - the evening (night watchman Tour) and walking the town walls in the early morning.

Trains serve all of these places well. You don't need a car. And if you are arriving on an overseas flight, driving while jet lagged is usually discouraged by most of us here.

You don't delicately say if you are including your arrival day in this itinerary so I will presume you are not. My recommendation, rent a car and spend the first night somewhere close to Munich's airport such as Dachau. Then, after a relaxing night's sleep, drive to Fussen and visit Hohenschwangau to see the castles. Remember, admittance to the castles are via a time-reservation only so allow time to stand in line for a reserved entrance ticket (at least 90 minutes later). You can reserve in advance via their web site (google Hohenschwangau ticket office) a minimum of two days prior to your arrival.

Then, the next day you can proceed up the romantische strasse to Wurzburg stopping in Nordlingen, Dinkelsbuhl and/or Rothenburg o.d. Tauber along the way.