Where to spend two extra days in Germany and/or Austria?

However I'd do it something like this so you are not losing time backtracking :

Fly into Frankfurt if possible go straight on the train to Rothernburg ob der Tauber from there.
From Rothernburg ob der Tauber train to Munich.
From Munich train to Salzburg.
From Salzburg train to Hallstatt
From Hallstatt train to Vienna
Fly home from Vienna

You might cut out something so you are not trying to do too much with the little time you have.

Is one of your extra days your arrival morning in Munich (3) and/ or your departure day from Vienna(3)? If so, I’d add an extra day to these cities.

Good point.

One of the three days in Munich is the day we arrive (likely early in the morning from the US). However, the day we are leaving Vienna is not counted in the 3 days in Vienna.

You will have a great trip.
Just an fyi -
I would check for holidays and which locations you'll be located on Sundays , as many stores & restaurants in Europe close on Sundays. I've find it's more of an issue in the smaller towns.
Safe - and healthy- travels!

If you're an art lover, then definitely add an extra day to Vienna.

I love Rothenburg, and have been there twice, and do recommend going. However, it is a 3+ hour train from Frankfurt, [did the trip this past December,] which involves around three train connections. This requires a degree of alertness and ease with navigating the German train system, after jet lag. Not sure where you all are coming from, but if you do fly into Frankfurt, why not spend the rest of the day in Frankfurt, before moving on? There are hotels actually inside the Frankfurt airport.
I'd also recommend the Rothenburg Night Watchman tour, and you'd want to be awake for that.
Aww, writing about this makes me long for the days when we can travel again!

We liked our first 5 day trip to Vienna so much that we went back the next year for a week, and still didn’t do all we had hoped for. 3 days is just surface scratching. One nice day trip is to Melk, boat in one direction, train the other.

Also, check out the dates for Octoberfest- do you want to join in, or avoid it? As it was canceled for 2020, I'm sure it will be extremely crowded in 2021.

Another thought I had was possibly joining an organized day trip from Munich to Rothenburg, but then we'd only see it during the day when it was crowded. I have read that it is beautiful at night, so I don't know if I like this option as much. I feel like we are neglecting Germany a bit is we only visit Munich, but we have limited time in the area. If only travel was free and our time unlimited! :)

"LOVE the mountains,"

You are missing the mountains with the options you mention! Add a night to Munich and head out to MITTENWALD. Take the Karwendelbahn lift, visit the Leutaschklamm gorge, see the violin/folk museum...

https://www.butterandfly.net/wp-content/uploads/Mittenwald-Germany-14.jpg

https://hastingshouse.typepad.com/hastings_house_us/2007/04/mittenwald_germ.html

Direct trains from Munich take you there in under 2 hours. 2 adults = €31 total round trip; buy the "Regio-ticket Werdenfels" day pass at the Munich train station.

Thank you, Russ. Mittenwald was not even on my radar, but it looks gorgeous. :)

I find it's best to plan by how many nights you will be in each place. For example 2 nights = 1 full day, 3 nights = 2 full days, and so on. That will give you a better idea of how much time you'll be in each location. Don't count the day you arrive in Europe and the day you leave as useful days. Also, don't underestimate the amount of time it takes to relocate from one city to another. Rothenburg ob der Tauber and Hallstatt are both doable for one night each to get a feeling for them without crowds.

How many total nights do you antipathetic being in Europe?

Bavaria is the last Bündesland for summer school vacation ending 13/09/2021. That said it will not be as busy starting 14 September 2021. No telling what the coronavirus will do to the school schedules in the future though.

I like Jazz's recommendations.
However, I loved the Romantic Road. There is more to the Romantic Road than Rothenburg ob der Tauber. It runs from Wurzburg to Garmisch.

https://www.romanticroadgermany.com

I agree with skipping Rothenburg. Its nice, but not worth that amount of effort and the time will be better spent in any of the other destinations on your list. .

If landing in Munich, I would forget R’burg and add both days to Salzburg.

if landing in Frankfurt, I would take 2 nights in Rothenburg

The places you have listed as visiting are big cities. I would certainly add either or both Hallstatt or Rothenburg so you can experience a small town.

If you love the outdoors, Ramsau am Dachstein is Amazing to hike around!

All the best, Monte