As for the rentals: I don't rent that frequently in Germany, but I think it's a case of YMMV. For one-day rentals, recently, I've seen Hertz offering nothing but 400km/day and Avis & Sixt 300km/day, but for multi-day rentals, you still can get unlimited mileage more often than not. Even in case you have Hertz status, it might be worthwhile to compare against Sixt, as they are the biggest player in Germany, I think. In the German market, Avis is probably bigger than Hertz as well.

As for recommendations on what to do, you don't have much time. I think Frankfurt is much better than its reputation on FT and well worth spending a day or two. Great museums, historically significant buildings and architecture, nice restaurants, good nightlife, decent daytime outdoor events during the summer. But it totally depends on your interests. Going to the Schwarzwald (Black Forest) for some biking, hiking, and such is an option (small detour on your way to Stuttgart/Goppingen). You're passing by Heidelberg on your way to Stuttgart, so that would certainly be easy driving-wise. Bergstrasse-Odenwald is an alternative to the Black Forest (and closer to your destination as well). I think I prefer the Black Forest for the dining options and cultural highlights in places like Baden-Baden, but Bergstraße is nice, too. It's also a wine-growing area.

TBH, I'm not a fan of Stuttgart and surroundings and hate the local accent. But if you're into cars, there exist some notable options there as well (e.g., Porsche-Museum, Mercedes Benz Museum). Also, you got some castles, old churches, and such. Dining in Southern Germany is usually also pretty decent. But generally, the stereotype of Swabian people is that they are hard-working, stingy, tidy, conservative, and close to nature. The Frankfurt area is a tad more liberal, alternative, open-minded.

Again, you really have little time. It boils down to your preferences.