Day 10: Rent car, drive to Nuremberg with stop in Regensburg for lunch/sightseeing
Day 11: Nuremberg Durer’s house and Nazi Documentation center
Day 12: Day trip to Rothenberg or Bamberg
Day 13: Drive to Mainz with stop in Wurzburg for lunch/sightseeing
Day 14: Rhine River cruise to Koblenz, stop and tour a castle
Day 15: Train to FRA and fly home
Day 11: Doc center is outside the center and somewhat time consuming, so you don't really have much time for Nuremberg at all. I'm surprised that the only sight in town you've officially targeted is the Dürerhaus. The science/tech museums alone look promising to someone with your stated interests. What about a walking tour? So I'd suggest that on Day 10 you add to your Nuremberg time by heading straight to Nuremberg rather than detouring to Regensburg, which in some respects duplicates the Bamberg experience on Day 12.
Day 12: Rothenburg will be the choice of most Rick Steves fans (most of whom are unfamiliar with and have never visited Bamberg - which I think means it's not really a "choice" for those folks.) But I will advocate strongly for Bamberg. As he's prone to do with his favorites, Rick Steves oversells Rothenburg. For those with 5 days in Germany, the only towns mentioned in his planning guide are Munich and Rothenburg... and for him, it's Germany's "best-preserved" (his words) old town, despite the fact that 40% of it was bombed out about 75 years ago and later rebuilt. (And here I thought "preserved" referred to "well-kept original stuff.")
Not saying you shouldn't go to Rothenburg - people do find it fun - only that you should understand the difference. Bamberg is a living, breathing university town where people lead all walks of life; Rothenburg relies almost solely on tourism for its sense of purpose - the vast majority of working folks there serve tourists in one way or another. Rothenburg is marketed and "cutesified" and largely overrun by tourist shops and international visitors; Bamberg gets visitors, but they don't overwhelm the place.
Rothenburg:
https://static.geo.de/bilder/e2/60/40947/article_image_big/2383b623e79431c7bd617551b69245a4.jpg
https://www.frankenfernsehen.tv/storage/thumbs/1200x630c/r:1475156718/95171.jpg
In a moment of complete honesty, Rick actually uses the words "medieval amusement part" in reference to Rothenburg. That's how it felt to me. And as far as the war goes, while Bamberg did not fully escape destruction, it did so better than Rothenburg. So yes, there's something to the UNESCO designation if you're interested in more than just stuff like the "implements of torture" museum.
If you enjoy beer at all - that alone is a good reason to choose Bamberg:
https://www.europeanbeerguide.net/bambpubs.htm
Mainz: I guess your spending 2 nights there, though you aren't really seeing Mainz. To keep 2 nights in one spot, I would urge you to choose one of the downstream towns instead. If you need to be as close as possible for a direct train ride to FRA on Day 15, Bingen is a good scenic-area choice.
CAR: I don't understand the purpose. It seems to be getting in your way. The car alone won't really work for your cruise. You don't need it for any of the towns you are visiting - you can easily do a stopover in Würzburg by train as well. You probably aren't sure HOW to do the trains for this part but I assure you it's doable and that we can help you do it.
CRUISE: Cruising all the way to Koblenz is inconvenient, expensive, and wastes good sightseeing time. Better to end the cruise in St. Goar or Boppard.