When in spring? Usually later is better, I tend to recommend May and ideally later in May.
Some of the recommendations depend on where you're flying into. For example, if you're flying into Germany, make sure that the rental car company will allow you to take the car into Poland if you want to go to Auschwitz. Not all rental car companies do allow you to take cars to all European countries, especially in an easterly direction.
For any travel between larger cities in Western Europe, my preference these days is the train, it's usually faster and - with the current fuel prices over there - cheaper. Plus you don't have to worry about parking the train, as parking in a lot of the larger cities in Europe plain blows.
The Nurburgring is pretty close to the middle of nowhere as Western Germany goes, so you'd have to get a rental car to go there. Plus another rental car to drive on it as most regular rental car insurance doesn't allow for that. Not that people don't do it, but things get ugly expensive quickly if something goes wrong. There are several companies that rent cars at the 'ring.
As to "where else" - I think that depends a lot on your interests. In Germany, the Auto & Technikmuseum in Sinsheim would be somewhere I'd definitely recommend, and it's something like an hour depending on traffic from the Porsche museum.
A lot of it depends on how much you want to travel - in general, the distances are much shorter compared to the US and Canada. The old joke about 100 years being a long time ago in the US and 100 miles being far away in Europe is definitely true.
For flights, you probably get the best choice/value combo if you plan on the international leg being out of NYC if you're flying from the US. I personally strongly prefer non-stop flights these days as there is slightly less scope for chaos. For example, I just visited family in Hamburg and instead of flying into Hamburg, flew into Berlin and took the ICE (high speed train) to Hamburg. Helps with the "passenger in Hamburg, bags in Instambul" type problems.
Other car things I can think off the top of my head would be to visit the Cite de l'Automobile in Mulhouse, France. That's the old Schlumpf collection with some other cars added on and the location where my avatar hails from. You'll be pretty Bugatti'd out after that. Mulhouse isn't that far from Stuttgart (about 3h).