I just did a personal ring trip by myself. I spent a week in Germany with 2 days at the ring. Round trip tickets were $800 from San Francisco. I spent about $40-50 a day on hotels/bnb costs. Rented a 2017 BMW 320D sport package 6MT for $40 a day unlimited mileage.
As for the actual ring experience, I did a dozen ring laps. I did two laps in a rented BMW M2 from one of the "rent for the ring" rental companies. They had prices that were cheaper than the rental costs from Apex. No cool Misha to instruct though - I got a Russian girl as an instructor.
The rest of my laps, I did in my regular rental 320D. I would say you can drive a regular rental car, but to do so with caution. IF anything were to happen, it would be very bad and you may not be covered under insurance and rental policies.
I have heard of rental companies blacklisting people, but that is more common if you rent a high dollar sports car. From what others at the ring have said, my rental diesel is not really on their radar of cars they'd track. If you rent a M/AMG/RS/Porsche, they will have trackers on those cars for sure. Again be cautious and do some research. If you do ask them, the official word is that they will not allow you to drive them at the ring.
Overall, I spent $2k on the whole trip including airfare, rental costs etc. The rest of the time I drove around Germany, visited the car museums, did a factory tour at Porsche and BMW, went sightseeing, drove the epic black forest mountain roads etc.
Oh, and I ran into Schmee of Youtube fame at the ring. Rather interesting guy.
Here's a lap with instructor in the M2 I rented.