Then I would take public transportation to an airport in Germany near Nurburgring (either Cologne or Frankfurt I believe)>

Just take train over border and pick up car in say Aachen - which is also very close to the Nurburgring I believe. And Aachen is again a very historic city to spend a few hours or night in.

I was just giving examples. I haven't even started studying the train routes yet to know where I can go to get a rental car in a non-airport area.