Suggestions with Germany and switzerland itinerary

Maximize your time in the BO. Mountain weather can be moody, so give yourself time if you have a bad weather day. As much as I love Lucerne, I would take a night from Lucerne and add it to the BO. No one is ever ready to leave. It's hard to describe the incredible beauty of the Alps, and the thrill of riding those gondolas to such altitudes! You have come such a long way, so spend it IN the Alps. Plus, you will have been on a pretty hectic pace, so it's time to slow down and breathe and just enjoy being there.
I haven't done the train route from Stuttgart, so can't comment on that aspect.
Lucerne - go to Mt Rigi, via the boat ride, and see the sunset at Mt Stanserhorn, after an open air Cabrio cable car ride.
It's an easy commute to the Zurich airport from Lucerne. The train goes straight into the airport. Swiss trains are the gold standard for dependability.
If possible, can you add a day or two? You won't regret it.
Have a wonderful family vacation!

Tip: also Munich has a car museum: the BMW museum and at opposite side the BMW Welt.

Journey planner for train connection (ensure to choose the right Wengen): https://int.bahn.de/en

The first two question will likely get better answers in Switzerland forum.

Are there any other ideas I should consider?

The Stuttgart museums seem quite important to you - I see no other purpose for the large detour to Stuttgart in your post. So here are some thoughts about your time in Germany that assume Stuttgart remains in your plans...

Day 5 Drop off car in stuttgart and take train to Wengen Switzerland.
Is this double?

This is a six-hour journey with 3-5 changes of train. It's not ideal.

The Romantic Road is only one of the options for seeing old-world Germany. You could instead do your exploring of Germany's old-world towns, castles, etc. between Stuttgart and the Swiss border. This part of Germany contains better driving scenery and many more castle options than the Romantic Road. Please see a list (with links) for destinations not too far from Stuttgart in this thread:

https://community.ricksteves.com/travel-forum/germany/travelling-to-the-black-forest-early-april-need-some-advice

(Please note that there are 30 town listings in the "Towns on the German Half-timbered-House Road" link.)

Some additional suggestions for you...

Wasserschloss Glatt (near Sulz)
Triberg (See Rick Steves)
Gutach and the Black Forest open-air museum + summer bobsled
Villingen
Löffingen

It's my impression that you are flying into Munich without any specific goals in Munich. Perhaps you should just fly into STUTTGART, rent your car there and tour the area between Stuttgart and the Swiss border... FREIBURG (a nice place to stay/visit) might be a good German city for dropping the car - you can catch a train from there into Switzerland and proceed to the Bernese Oberland. This transport from Freiburg > Wengen should take less than 4 hours.

Or perhaps you fly into ZRH - The train ride from Zurich to Stuttgart is only 3 hours - and out of ZRH at the end as well. My choice for my first day/night after landing at ZRH would be to catch the first train to Schaffhausen, a beautiful old Swiss town near the German border and only 35-45 minutes by train from Zurich. It's also near the Rheinfall (a most impressive waterfall) and right on the train route to Stuttgart, which makes it a bit of a no-brainer as a stopover. (Note that if you were flying into Munich, the train ride from MUC airport > Munich would require roughly the same amount of time as this train ride to Schaffhausen - so with Schaffhausen, you'll still get to your Day 1 destination quickly enough.)

Luzern is a better choice than Zurich

Thank you so much for all the responses and suggestions.

I have never been to Germany or Switzerland so not sure how far each town are.
Lot of this information is very useful.
Trying to figure out the iterinary that makes my husband and teenager happy so I might be thinking in multiple directions.
I will check the options to fly to stuttgart or Zurich instead of Munich as well.

We lived in Augsburg for four years and have visited all the places you. mentioned.
Unless you are a fanatic about cars, skip Stuttgart.
Glad you plan the Romantic Road, it is special, you can't do it in one day. If you start in Fussen, you will need to plan at least two days, and a full day for Rothenburg ob der Tauber.
Augsburg is worth half a day (visit the downtown, Rathaus and Fuggeri).
Lucerne is great, skip Zurich.

Thanks for the reply.
My son is fan of the sports cars, so I thought the porcha and Mercedes museum might be interesting for him. Is the BMW museum near Munich good option well?