Autos:
Stuttgart - Mercedes-Benz (Mercedesstraße 1!)
Affalterbach - Mercedes-AMG
Zuffenhausen - Porsche
München - BMW
Ingolstadt - Audi
Wolfsburg - VW AG
Sinsheim - very popular tech museum with all sorts of transportation, planes, military stuff, cars, etc
Berlin - why not just take a flight from Frankfurt to the capital if there is free time for that, awesome place
Saarland - not that far away from Darmstadt, a bit of French flair there, great country side, Saarbrücken and Trier are nice historic places and Rally Deutschland is held there in August
I can tell you from experience (been there twice) that the MB Museum will take you almost all whole day, so keep that in mind. The Porsche one is smaller, but you shouldn't miss the showroom that is very near the museum so that won't take you just a few hours either.
BMW Welt & Museum you can do in one day easily if you arrive there in morning. Don't forget the city though - very nice city centre (also good for shopping but that applies to all larger German cities); Olympiastadion and Allianz Arena worth visiting too (there's a BMW Niederlassung dealership very near the Arena if you can't have enough Bimmers), huge Merc Niederlassung dealership not far away from BMW (click). Sadly, I haven't been to the centre of Stuttgart but go there too as you really can't make a mistake in Germany.
Audi Forum in Ingolstadt is quite forgotten by people, but it is also worth visiting. The whole complex is very nice and modern (feels a bit like you're in an Audi actually :p), their restaurant makes some great food too.
Ad Nürburgring rentals: Rent4Ring offers rental cars as well as RSR Nürburgring - this company also offers Taxifahrten, you can pay for a lap with a pro driver. BMW has their own service, too, but it is more expensive than the other companies IIRC.
Look up BMW On Demand - BMW's own rental service with not so bad pricing. if you're old enough, you can get an M car.
That's all I can think of at the moment. :mhmm: