I understand now, that the static friction opposes the tendency of a surface to slip over another.
Exactly! Let's remember this point in the next sentences.
When a wheel rotates, static friction pushes the wheel forward to keep the contact surface stationary with respect to the ground.
Not necessarily forward. It can also be backwards. It pulls in whatever direction necessary to keep the contact point stationary. And only if necessary (e.g. there is no static friction when a wheel rolls at constant speed over a horizontal surface. There are no forces at all present, so nothing for the static friction to oppose. The rolling motion just continues effortless until stopped.)
When a car is going at constant speed down a hill, static friction must push the car down the hill where you also have gravity helping too.
No. It pulls uphill.
Think for a moment of a star rolling down. It's legs touches the ground one at a time. While a leg touches the ground, it must not slide. We can think of it as a stationary object in that moment while it is touching. Downwards forces (gravity) must be opposed by static friction upwards, just like for a stationary object resting on the hill.
Then the next leg takes over, and the same thing is the case. Static friction must hold back upwards to avoid sliding of the leg.
Now add more legs to the star. Many, many more. Soon you almost have a round circle, where the legs are the "points" of the circle that touch the ground for only a split-second. Nevertheless, the same is the case; while touching, static friction holds on to the touching point upwards to avoid it from sliding because of gravity.
[...] for the car to go up the hill, where the static friction is again pushing the car forward, countering the slipping of the tires that would result due to the rotation of the tires.
Yes. Again static friction pulls uphill to prevent sliding because gravity pulls down. The direction of static friction does not depend on the rolling direction; it doesn't care if you roll up or down the hill.
If the wheel accelerates at the same time, the static friction might point differently. This is again regardless of the rolling direction but only the acceleration direction comes into play.
How can you get a constant velocity when going down hill, when the static friction and gravity are both creating a net force down the hill (if you ignore rolling friction)?
You are exactly on point here. It can't have constant speed, if all forces pull the same way. Such thinking will lead to the understanding that static friction in fact must point the other way.
A question asks me to find the steepest hill a car can descend at constant speed given the static friction coefficient, but I think what is needed is the rolling friction coefficient. I don't think its possible to answer the question without rolling friction.
As mentioned in a comment, most questions would assume ideal world-models. No deformation of surfaces e.g. So rolling friction will be assumed 0 most often, unless you drive on a clearly non-ideal surface, like a sandy beach or a flexible trampoline.
The idea to solve this question is to remember the formula for maximum static friction:
$$f_s\leq \mu_s n$$
If you have a certain friction coefficient $\mu_s$, you can do your Newtons 2nd law calculations on the car and put this formula in with an $=$ instead of $\leq$, because you are looking for the maximum limit. Then solve it for the slope angle (the angle will be a part of the force components).
Forget about pushing friction and slowing friction. Think of static friction and kinetic friction.
Static friction is friction between two or more solid objects that are not moving relative to each other. It's what keeps the car from slipping. When the car is in motion, ideally, the tyre and road surface do not move with respect to one another, the tyre grabs the road. It works the same way with the soles of your shoes and ground.
Work done is equal to Force times Distance. Since the tyre and road are not moving with respect to each other, no work is done against static friction, nor can it ever. When you are cruising at a constant speed on a level road, the engine is working against friction, but this is kinetic friction: the friction between internal parts of the car's engine and drivetrain and the friction between the car's body and the air.
There is also loss within the tyre as it rotates. The tyre flexes as different sections of the tyre come into contact with the road during rotation. The deformation is not perfectly elastic and some of the energy is lost as heat during the process. Underinflated tyres can add to the effect and increase fuel consumption. Recommended tyre pressure is a trade off between comfort and handling.
When you apply the brakes, they are designed to cause kinetic friction between the brake components (pads and rotors for disk brakes, shoes and drums for conventional) which converts the kinetic energy of the car into heat. Electrical cars can convert some of the kinetic energy back to electrical energy which is a more efficient use of the kinetic energy.
When the tyres slide, as when you go into a skid, kinetic friction between the tyre and the road does slow the car down, but nowhere nearly as efficiently as the brakes would, which is why modern cars have anti-lock brakes. Besides, steering is nil when in a skid.
Best Answer
If you're considering an idealized situation with no rolling resistance, then once the wheels are rolling with uniform velocity/angular velocity such that $v = rw$, the force due to static friction on the wheels would become zero since the speed of the point of contact would be zero (principle of superposition)