Interests include nature walks, museums, castles, biking along the
Rhine, scenic drives, cafes, city sights, kid-friendly spots.
Thoughts...
After Berlin - I see big-name places like Munich, Frankfurt, Strasbourg and Stuttgart on your list, but not much for the scenery, the nature walks, the biking, the castles, or the kid-friendly spots that are also on your list.
Generally speaking, driving within Berlin and into Munich, Stuttgart, and Frankfurt should be avoided. And since kids, IME, find train travel a lot more interesting than riding in the back seat - or even in the front seat - I am even more puzzled that a rental car is your choice.
Biking on the Rhine (and its tributaries, like the Mosel) can be a nice experience. But the Rhine Gorge isn't on your itinerary. It really should be if you are interested in castles (which aren't mentioned in your itinerary or implied by your destinations.)
Suggested itinerary:
Day 4) Instead of 1 night in Munich, stay in Rothenburg.There's not much for kids here, and it's a small town. Some afternoon and evening hours on Day 4, maybe a couple of morning hours on Day 5.
5-8) Salzburg: you need at least 3 nights here. On Day 5 I suggest you visit REGENSBURG on the way to Salzburg. It is a UNESCO World Heritage Site and Germany's best-preserved medieval city. Only Rick Steves can explain why it's not in his book.
Salzburg... More interesting than the Salt Mines IMO are these sights, a little ways from Salzburg up the Salzach River, in the town of WERFEN.
-Hohenwerfen Castle and Falconry show
-The Eisriesenwelt Ice Cave
-The Sound of Music hiking trail
Maybe some time in the Berchtesgaden area? It's very close to Salzburg. There are salt mines there too, but also Lake Koenigssee and great alpine scenery from the Jennerbahn lift.
9-11) Black Forest... Skip Freiburg and Baden-Baden if you want nature walks, scenic train or car rides, etc.
These attractions are in Gutach:
GENGENBACH is a wonderful old walled town and a good travel base if you want to stay in if you want to visit Gutach, Triberg (waterfall, Black Forest Museum), Schiltach, and Strasbourg (FR)
12-13) Stuttgart area... car museums... I'd look into Esslingen, just outside Stuttgart, as a pleasant place to stay.
14-16) Rhine Gorge/Middle Rhine Valley / Mosel River Valley.
Biking/hikiing or easy walks
River cruising
Burg Eltz Castle, Marksburg Castle, Rheinfels Castle
Old-world towns to: Oberwesel, Bacharach, Cochem, Braubach, Rüdesheim
Base town: Boppard is excellent... many hotels and restaurants chairlift ride there, easier access to the Mosel and to Burg Eltz than from other Rhine towns. Free area train rides come with your room or apt.
17) Final day/night in Mainz or Frankfurt