Ok, after some study I found this:
1 - The amount of driving or damage does not affect the amount of money to be paid. As soon as you have driven a car you have the option to refresh the rigidity and it will be for a fixed price.
2 - The rigidity fix of all cars bought at the dealerships will cost half the price of the car (Volvo C30, price: 42k, fix: 21k; M3 coupe, price: 92k, fix: 46k
3 - All chromeline cars can be fixed for 500 credits, probably a friendly gesture to all who bought special editions.
4 - There is no indication yet about whether or not (or at which point) the lack of body rigidity will become an issue. But you probably don't have to refresh before you've driven a few k of miles.
5 - I'm not yet sure what the price is for second hand cars, but I'm sure that's related to the price of the car aswell.
edit: I don't know how people came to think it was related to damage. I haven't found any indication of this as of yet.