Question: I've been doing physics problems regarding cars for a while. I understand that there is a static friction (which appears when the wheel is rolling) and kinetic friction (which appears when the wheel is sliding). However, the way I'm visualizing it, static friction between the tire and road should not stop a car. In fact, when I asked this question to my teacher a long time ago, he said that it is actually the friction between the axle and the wheel that stops the car, and the road friction actually helps the car to move. But I know that when the car slips, the friction decreases and thus stopping time increases. How could this possible be linked to the axle? What is going on?

Answer: 
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> the way I'm visualizing it, static friction between the tire and road should not stop a car.
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Static friction is able to supply a force. If that force is opposite the direction of motion, it is able to stop the car. In the case of your car and the brakes, that's exactly what happens.

A torque from the brakes is applied to the wheel. This torque becomes a force against the ground. As long as the force is not too great, the wheel doesn't slip and the road supplies a force back on the wheel (which slows the vehicle).

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> I don't think the vehicle is slowed down by the static friction between the wheel and road. 
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And yet it is. To see that this is true, let's imagine a situation where we remove static friction. Drive the car onto a patch of ice where we assume the coefficient of friction drops to zero. The car continues to drive at the same speed.

When we stomp on the brakes now, a torque is still applied to the wheel, but now the force of static friction is zero. The car does not slow down and continues at the same speed. Only when (hopefully static) friction is present can we slow the car.

Kinetic friction is also capable of slowing the car, but we don't want that because we don't want to skid the tires.

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> ...that should be the static friction between the car and the road
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The only portion of the car that touches the road is the wheels/tires. That is the only part where friction can develop. So to say the friction between the car and the road or the friction between the wheels and the road is the same thing.
Have you ever tried to lift yourself up by pulling your own hair? Go ahead try it. Try it really hard. You should be able to levitate a couple of inches, right? What? You got a handful of hairs in your hand and a bald patch in your skull? It should serve you right. Hopefully, you will learn that internal forces cannot change the total momentum.

The friction of the gears, engine, etc. are internal to the car. They cannot change the momentum of the car. For all we care, replace the gears, engine, and breaks with a couple elves and other mythical creatures (a.k.a. hidden variables). As long as those hidden variables are internal to the system (car) they cannot change the momentum of the car.

If you insist that friction cannot stop the car, I dare you to drive really fast on an icy road where there is almost no friction. Hooray for Darwinism.