As i got there almost weekly when the sun is around, I can give you some answers.
Best days to go for a free track : during the week, or Fridays, even though most Saturdays are still quiet.
Try to avoid the big holiday weekends, Easter, summer, etc, as there will be a lot of your fellas from Uk going down to the Ring at the same time, and that means everybody absolutely wants to make the weekend worth it, everybody drives like nuts, most crash, some die, and the weekend turns to shit.
Doing 4 laps in a day is easy, as long as you don't spend 3hrs wandering around on the parking lot (which I can understand given the cars which are there!) between every lap.
On a big day, I achieve to cover 27 laps, juste to give you an idea.
The cost of the laps, you know. Inculde some food, fuel, boat, etc for the trip, maybe a hotel (gastehaus) around the Ring.
Have a look if your insurance includes tracks, but anyway the Nurburgring is classified as a toll road, and that's what it says on the paper if you crash, not "closed racetrack". So in case of a crash, the insurance might be a bit of a handful but they will have to cover you anyway as it is a Tourist Day after all. That doesn't apply for organised trackdays where you are officially on a closed racetrack. BIG difference here.
Well, That's what I can tell you so far, if you have questions, don't hesitate.