"We'll be visiting Bavaria for about a week as part of a longer trip, and would like to see Fussen, Rothenburg, Nuremberg and Salzburg. We're considering two options: first, staying in Munich and day-tripping by train; or staying just outside of Munich, maybe in a castle converted to a hotel, and renting a car."
Daytripping by train from Munich to these other places you name isn't very convenient at all, IMO., since...
1.) you'll spend 4 hours on the train to Füssen and back (5 if you are going to the palaces in Schwangau)
2.) you'll spend 5.5-6 hours getting to Rothenburg and back - on the fastest, most expensive trains, 6.5-7 hrs. on local trains.
3.) daytrips to Nuremberg will take only 2.5 hours round trip on the fast trains but normal fare is about 200€ round trip for two.
4.) daytrip to Salzburg is 4 hours round trip on local trains, 3 on the fast ones.
By car, your travel times will also be very substantial. Your destinations are pretty far apart for daytrips, and all this travel time adds up.
Suggest you split your time in 3 different places. This reduces your total time on trains substantially and you can use inexpensive daypasses (which use the local trains.) Example:
Rothenburg, 2-3 nights: Daytrip to Nuremberg and back (1.25 hrs each way) for two passengers is 16.80€ total with the Tagesticket Plus daypass. Alternatively, overnight in Nuremberg and daytrip to Füssen, which makes the subsequent trip to Füssen shorter by maybe 30 minutes.
Füssen: 2 nights. Get there on a Bayern ticket (26€) using local trains (the only ones you can use between Munich and Füssen anyway.) 4.25 from Nuremberg, 4.75 hours + from Rothenburg.
Salzburg: 2 nights. Get there on a Bayern ticket (26€) again. Around 4 hours.
That's about 70€ altogether, and around 11 hours of train travel (vs. around 20 hours doing daytrips from Munich.)
I'd try to do the either the trip to Füssen or to Salzburg on Sat or Sun so you can use the Bayern ticket prior to 9 am. The other option is to advance-purchase a savings fare ticket for these trips for tickets that include the faster trains. You'll save maybe an hour on one trip and half an hour on the other, but you'll have to stick to a schedule as these tickets are train-specific. Fares will be close to the Bayern ticket price if you get tickets right when they go on sale (92 days out.)
2-base Option w/ MUNICH as 2nd base: Instead of staying in Füssen and Salzburg you can use Munich as a base, but the daytrips are still long...
Daytrip to Füssen: bite the bullet, get up very early, go to bed very late. Go on Sat or Sun to use the Bayern ticket early in the am.
Daytrip to Salzburg: 2 hours each way on RE trains, Bayern Ticket. Saturday or Sunday again so you can leave early on the Bayern ticket.
If you stay outside Munich in a small town instead of Munich, you will make the train trip longer to either Salzburg, Füssen, or both.