The answer is yes to all points you raised. To support the weight of a vehicle with a reasonable amount of air pressure, the tire needs a certain amount of contact area with the road. In this sense, the weight of the vehicle determines the overall size of the tire, which is why large trucks have large tires (and relatively high inflation pressures) and small cars have smaller tires.
Sports cars require minimization of unsprung mass in their suspension systems to maintain good contact with the road surface at all times, and lots of surface area in contact with the road which requires minimization of tire mass and maximization of contact area. This is accomplished by making the tire as wide and thin as possible which results in a tire that resembles a thin belt of rubber wrapped around the rim of the wheel.
The speed at which the tire is intended to roll also affects its design. High speeds require careful manufacture to prevent the tire from being out-of-round or unbalanced because these things would make the tire vibrate badly at high speed. In addition, as the tire rolls it flexes and thereby generates frictional heat within its structure which must be dissipated or else the tire will soften from the heat and either explode or catch fire. Tires designed for high speeds must therefore be made of more heat-resistant rubber than those intended for ordinary use.
The intended use of the tire determines the tread pattern, with race car tires having extremely soft surfaces with no treads in them at all and tires for use in snow and ice having very deep and open tread patterns and sharp tungsten carbide studs inserted into the tread surfaces.