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the weight thing is mostly a function of air weighing less than aluminum, even pressurized air in tires weighs less than going to a biger wheel and less tire then you have to make the wheel stronger. A lot of this has to do with the type of chassis. In F1, its probably somewhat restricted by rules (im not sure i'll look) but a lot of it comes down to the tires acting as the springs. the more sidewall the more spring. I can tell you for sure that Bridgestone and Michelin are playing with that tire to tire race to race. The function on a road car is that of reducing that sprign when you go to a smaller profile tire. this stiffens up the suspension and the handling as the tire isn't left to wander around. The dragsters you mention actually use this sidewall flex to the extreme. If you watch superslowmoion replays of drag launches, then you can actually see the sidewall winding up and then releasing (cthey run 2-5psi) this creates some forgiveness in the launch and helps to prevent wheelspin in that case
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