I'd like to take my car on a couple laps around the Green Hell (Nordschleife track). I have no experience as a race driver. I'd rather not crash my car but I don't want to go slowpoke speed either. Actually, I'm a little scared and I want to do this to increase my confidence, so I'd like to get some feedback while driving. The FAQ and various internet sources suggests that hiring an instructor is possible, but I cannot find where and how. (How) Can I hire an (English speaking) instructor without also renting a car?
I think what you have exactly in mind is not really possible. Driver instructors which are doing their job for a living need to have insurance and they will lose it if an accident happens and, even worse, could be held responsible for the accident. This is an inacceptable risk.
There are many Motorsport freaks driving on the Nürburgring who probably could give you instructions for free, but this is like lottery and I don't recommend it.
On the other hand, driving your own car is possible, but its called "Green Hell" because of trees (For the people who don't know what that mean: No straw bales, no cushioning if things go wrong). Too many dead people ending up on one, so I do not recommend trying to avoid slowpoke speed.
So either drive in your own car carefully or use an instructor with an insured car. Both are AFAIK not possible. Sorry for the misunderstanding.
The address for getting an instructor is the Drivers's Academy which allow to drive either a sports car or even a fully functional race car. They also explicitly mention support for beginners: