I think there's an important lesson for the OP lurking in this question.
Anything as large as a car wheel is almost certainly extremely complicated as well. The wheel has more atoms than a beach has grains of sand. It's made of long, convoluted polymer molecules that were formed in complicated chemical processes, as well as from air and metals, themselves interesting materials to understand. The wheel deforms and bounces as it rotates. It dissipates energy into the road and its surroundings, and takes energy in from the motor. If you wanted to understand everything about the wheel, by the time you are done, you'll have a learned a great deal from all over the natural sciences. I once read an entire book called "The Bicycle Wheel" that really only covered a few select topics, like the tension in the spokes and the alignment. Everything that can be said would take many books.
On the other hand, physicists do not want to know everything about the wheel. Part of the art of physics is figuring out the bare minimum amount of information about the wheel to take into consideration. What to consider depends on what sort of question you're answering.
Suppose you are testing the brakes, and you want the car to stop very suddenly. Then the brakes must dissipate all the kinetic energy of the car. In that case, you might wonder, "Can I calculate this energy using simply $KE = 1/2 mv^2$, or do I need to consider the extra energy of the rotating wheels?" Here we would ignore most sorts of energy, and focus only on the translational and rotational kinetic energies.
In another situation, we might ask, "How will my gas mileage improve if I inflate my tires another 5 psi?" (In the US, we use absurd units for everything.) In that case, we would be concerned with how the energy of the wheel is being turned into heat, and we'd have to consider the energy involved in deforming the wheel repeatedly as it goes around - stretching the rubber and slipping against the road, etc.
It's true that there are many forms of energy in the wheel, and so the full picture is complicated. However, the goal is to ignore as many complications as is reasonable at any given time. Figuring out which things to consider and which to ignore is a difficult skill that physicists gain only by dint of long and diligent practice.