Yup! Springs in series. If the suspension spring would appear to provide a wheel rate of 222.22 pounds per inch, but the tire has a rate of 2000 pounds per inch, actual wheel rate would be 200 pounds per inch.
As for damping. 'most everyone has dribbled (as in basketball) a tire and wheel assembly across the shop floor at some time or another, so we know there's not a whole lot of internal damping in the tire. Good thing, of course, or we'd be throwing treads from the heat buildup.
Perhaps someone closer to the sport can provide more information, but I heard rumors that, before rule changes to limit aero downforce, designers of Formula One cars were considering eliminating the chassis springs entirely. Don't know what International Karting Federation class they would have fallen into.