score:4
travel the entire Iceland by camping.
Is it possible and is it allowed?
Yes, as long as you stay on campgrounds. Iceland does not have a generic "right to roam" permission allowing anyone to camp almost anywhere, unlike Norway, Sweden, or Finland. The right to camp in the wild for a night only applies if you can't reach the next campground, which applies if you are on foot or by bicycle, but not when you are by car.
I witnessed people being fined for wild-camping with a car in the interior Iceland, so I asked about the rules, as I had been wild-camping on my hike for more than a week. That's how I learned about how those rules work in practice. You don't want to pay that fine if you are on a tight budget, so if you are by car, please stick to campgrounds.
The cheapest holiday in Iceland is probably one where you bring your bike, and camp alternating on campgrounds or in the wild. That or hitch-hiking, but hitch-hiking has its own downsides, in particular on roads with very low traffic (see also: How many cars/day pass over route F35 KjΓΆlur in September?). If you take the ferry, you might be able to take a fair amount of food too, saving more money (but the ferry is not cheap, certainly not if you bring a car).
Has anyone done this before?
Yes.
Can I still experience the culture, and meet people?
Yes. You could camp near a city and travel into the city to experience the nightlife before returning to the campground. Of course, cultural tourist attractions such as Γingvellir are reachable by bicycle as well.