Switzerland, Germany, Austria Itinerary

I think one night in Rothenburg odT is enough, you could add that night to Garmisch then.

"RodT looks lovely but should we skip it and spend more time in Garmisch? Or is it all worth the car time? "
Here are some of my thoughts...

It's not only the car time to consider but also the time spent...
- on the unpacking, packing routine (9 nights in 5 different places)
- in tourist-heavy destinations
- in the Alps, where the weather may or may not be pleasant in May

Also consider the Pfingsten holiday weekend (5/14-16) which will impact road travel and tourist destinations.

When you pin down which places you will visit on which days in advance, as you're doing, you have less flexibility to adjust your sightseeing to weather, holidays, special events, personal illness, etc. I usually prefer to locate a smaller number of towns to stay in for several days and use as travel bases - maybe 2 or 3 in your 9-day trip - and to do day trips from those towns to nearby places of interest. Just for example with some of the towns you have in mind...

Stay in Munich a few days, day trip to Füssen, Garmisch, Salzburg.
OR...
Stay in Prien (home of Ludwig's Herrenchiemsee Palace half way between Munich and Salzburg) and day trip one hour into Salzburg or Munich.

I do not happen to think Rothenburg or Neuschwanstein (which isn't a castle but a late 19th-century palace) are particularly worthwhile Bavarian destinations, in part because they're so touristy. I'd encourage you to look a little further into the possibilities. If you are going to be in the Rothenburg area for sure, there are some GREAT destinations and sights near there (Bamberg, Würzburg, Nuremberg, Bad Windsheim, Iphofen, Marktbreit.) There are also some genuine castles worth visiting in Germany (though not as many in Bavaria as in other places) including Burghausen near the Austrian border; check out Hohenwerfen (outside Salzburg) and Hohensalzburg in Salzburg too. The medieval city of Regensburg might be a good substitute for Rothenburg - and it's an easy day trip from Munich as well as a UNESCO World Heritage site.

Since you have a car, you can see the castles just as easily from Garmisch as Füssen.

Give Garmisch more than one night. Not because the town itself needs all that much time to explore, but if you want to ascend Zugspitze or do any hiking, you can't expect the weather to cooperate on any one day, especially in May. With a car, you can easily visit the castles near Füssen from here. Devote your first available day with clear weather to mountain excursions. Neuschwanstein wouldn't be ruined by cloudy weather in the same way that Zupspitze would.

Russ and I are of the same opinion about Rothenburg, but not about Neuschwanstein. I simply didn't find Rothenburg unique enough to justify all the attention it gets. It's a perfectly attractive and enjoyable town, but if you have to drive more than two hours out of your way to visit it, you've probably passed by any number of other acceptable alternatives. Unless you want a particularly high quota of souvenir stands.

However, Neuschwanstein, to me, fully deserves it's iconic status. Yes, there are massive amounts of tourists who visit, but where else will you find such a stunning castle/palace in such a magnificent setting? My only complaint is that the tour guides don't really do a great job in placing the structure in the appropriate neo-Romantic context, but with all the visitors they need to fit on the tours with only so many hours in the day, how much time do they really have to fully explain 19th century Germanic Romantic Nationalism and King Ludwig's obsession witht the operas of Richard Wagner? A little pre-reading, however, goes a long way.

Oh, and also... I'm commiting Ricknik heresy here, but Salzburg really didn't blow me away. I found the Altstadt surprisingly small and for the Austro-Bavarian corner of the German speaking world, surprisingly monochromatic. Take away the nostalgia many people feel for a certain immensely popular piece of musical cinema and the fact that it can claim Mozart as a native son, and I really think that Salzburg belongs in the same league as Landshut, Passau, Kelheim, Rosenheim, and Burghausen, but not Munich or Vienna.