Inertia is a concept—the property of an object to resist any change in its velocity. Momentum is a vector quantity—the product of an object’s mass times its velocity and an important characteristic of a moving object. Momentum is important because it’s conserved and it’s conserved in large part because of inertia and related concepts.
As a car heads forward, its freely turning wheels begin to rotate. The torque that starts them rotating comes from static friction with the ground. The ground pushes backward on the bottoms of the wheels to keep them from sliding and this backward frictional force exerts a torque on the wheels. They begin to rotate so that their bottom surfaces head backward and their top surfaces head forward.
The car’s powered wheels turn for a different reason: they are driven by a torque from the car’s engine. As you step on the accelerator, the engine exerts a torque on the wheels and they begin to turn. They would skid backward across the ground where it not for static friction between the wheels and the ground. This static friction opposes the skidding by exerting a forward force on the bottom surface of the wheels. This static frictional force produces a torque on the wheels and that torque partly balances the torque from the engine. The wheels don’t skid and they’re angular velocities increase relatively slowly. However, the forward frictional force on the wheel’s bottom surface isn’t balanced elsewhere in the car and the car experiences a forward net force. The car accelerates forward.
While it’s true that there is no thermal energy made by static friction, since the object doesn’t slide, your body can still make thermal energy directly. You get tired because your muscles must turn useful food energy into thermal energy whenever they are under tension. If you are doing work, they also convert food energy into that work, but even when they aren’t doing work, they still convert food energy into thermal energy.
This is an interesting question with several answers. First, a solid rubber tire would have a huge mass and would require consider work to accelerate. Because it rotates as the car moves, a tire stores twice as much kinetic energy as the other parts of the cars. By reducing the mass of the tires, the car reduces the amount of energy it must put into the tires to get them moving and the amount of energy it must remove from the tires to stop them from turning.
Secondly, a solid rubber tire would be so hard that it would give the car a very rough ride. The air in the tires cushions the car against many of the rough spots it drives over. Without the air cushion, the wheels and axles would bound up and down with every pebble in the road.
Lastly, a solid rubber tire would be very expensive. The materials used in a tire are expensive and a tire’s cost should be roughly proportional to its weight. Since a solid tire would weigh much more than an air-filled one, it would also cost much more. Its tread would still wear out, so it wouldn’t last any longer than an air-filled tire.
When you are pushing on something without doing any work, your energy is being converted directly into thermal energy inside your body. Your muscles are inefficient and they convert food energy into thermal energy whenever they are under tension. It’s like a car, which uses gasoline even when it’s stopped at the light. The engine keeps running but it does no work. Similarly, if you simply burned your cereal in your breakfast bowl, you would turn its energy directly into thermal energy without doing any useful work. Your body is also able to burn up that food energy and create thermal energy, albeit a little less visibly.
When you pull a wagon forward, friction from the ground starts the wheel turning and it does this by pushing backward on the bottom of the wheel. Friction is thus preventing the wheel from skidding across the pavement. When you step on a car’s accelerator, the car’s engine starts the wheel turning and friction from the ground pushes forward on the bottom of the wheel to prevent the wheel from skidding across the pavement. In the first case, friction is trying to help the wheel to turn while in the second case friction is trying to keep the wheel from turning. That’s why the forces (and the resulting torques) on the wheel are in opposite directions for the two cases.