Stuttgart and Munich. Help me fill the gaps

I would get in contact with Stuttgart Steps. They can plan a tour for you to cover Stuttgart, the cool palaces around this city as well as Esslingen, one of the beautiful, medieval towns in this part of Germany.
http://stuttgartsteps.com/
I like Heidelberg, but not sure if it is worth going out of your way to go see it.
There is an open-air museum in the Black Forest that would be nice for the girls to go see.

Hi,

I like both Heidelberg and Tübingen, famous university cities and would recommend visiting them.

Yes, Burg Hohenzollern in Hechingen, if you're into Prussian history.

Will you have a car in Germany? Hohenzollern can be visited by train and bus but it is a little awkward. Their excellent website gives details and they promptly answer emails.

Tubingen is an interesting, little visited by Americans city. If you go, take a ride on their version of gondola - stocherkahn. Get tickets at the excellent Tourist Information Office by the Neckar bridge.

If you visit Rothenburg, stay overnight. Take the Night Watchman tour

If you follow all the suggestions so far you'll run out of time. Lets just focus on a couple things; Rothenburg (RodT) and castles. Stuff you can easily do in 2-3 days that will be fun and help the girls get their Merit Badge.

RodT is not on a direct line between Stuttgart and Munich, but it's not hard to get to either. Heidelburg for example is in the opposite direction, as are some of the other recommendations. The Alps, and the extreme southern parts of Germany are likewise inconvenient for the travel time, because you want to spend time doing stuff, not traveling. So, let's make seeing RodT a priority.

If we do that, and look around where that takes us, we can find a number of excellent places to visit that are easy to travel too, and minimize transportation issues.

Here's my recommendation:

1. Plan to go to RodT. You can easily get there by train if you don't drive. (I recommend the Criminal Museum for kids that age.)

2. Spend a night in Colmberg Castle. That way your girls not only can say they saw a castle, but they slept in one. It's 20 minutes from RodT, and easy drive or cab ride. (You might ask if they have a shuttle.)

3. Go to Nuremburg. One of the very best cities to walk around, has a good Toy Museum (along with others), has a castle and walls, and all kinds of other attractions, and is an easy train ride into Munich. You can easy spend 2-3 days in Nuremburg alone, but even one is better than almost any other for sights and convenience.

4. Go to Munich. Spend some time walking the open air market, the English Gardens, the Deutsches Museum, etc. Unwind a little. Take the girls to a beer hall (yes, they can go in), give them stuff to talk about they can't do in the States.

I'll bet a trip like that will make them want to go back. Running for "must see" to "must see" gets old fast.

If you are coming from Paris Strasbourg is a wonderful stop off on the way to Stuttgart. 2 hours on the train. Strasbourg to Stuttgart is another 2 hours train trip.

Explore possible routes on Bahn.com

https://www.bahn.com/en/view/index.shtml

If you are busy in Stuttgart on September 7 and you have a hotel reservation in Munich for September 10 - for departing on the 11th - You actually have only 3 days/2 nights unallocated for Germany.

Both Rothenberg and Burg Hohenzollern are "out of your way."

Rothenberg ob der Tauber is north and will take nearly three hours on the train from Stuttgart so visiting there will take two full days of your time if you include an overnight.

Burg Hohenzollern is quite a ways south of Stuttgart, opposite to Rothenburg....

Another possibility is to visit Nuremberg, which also has a medieval castle, although a rebuilt one, and outstanding museums of many kinds.

https://museums.nuernberg.de/nuremberg-municipal-museums/

Stuttgart is 2 to 2 1/2 hours on the train to Nuremberg. And Nuremberg is an hour from Munich.

Good luck and have fun.