for both these questions, how do we frame the force equation and which direction does friction act in both cases?
If you look at the microscopic details, friction is complicated. But if you look at the forces, it can be described simply.
Static friction: If you press two objects like a tire and the road together, friction keeps them from sliding. If you push them sideways, friction is a force just strong enough to oppose the motion. This is like the reaction force the road exerts that hold the car up. It is just strong enough to keep the car from sinking into the road.
Dynamic friction: If you push harder, static friction is not strong enough to prevent motion. The tire slides. Friction is a force in the direction opposite of the motion. The magnitude of friction is proportional to the force pushing the tire and road together.
When a tire is rolling, static friction keeps the wheels from sliding. When a wheel is skidding, dynamic friction is acting.
For a rolling wheel, first imagine a wheel held above the road. It isn't rotating. It is being carried along the road. All parts of the wheel move at the same speed. Now add rotation. The bottom of the wheel has a backward component and the top a forward component. The total speed of each point of the wheel gets faster and slower as it goes from top to bottom and back. There is always some point at the bottom that momentarily has the slowest speed of all.
If the rotation is just right, the backward component of the bottom cancels the forward motion. In that case. The bottom point momentarily comes to a stop, matching the speed of the road. If you lower the wheel so that it touches the road, it is rolling.
For any object to move in a circle, there must be a force that deflects it from a straight line. This force is toward the center of the circle. It is called centripetal acceleration. For a car, it is static friction. If the car was on ice, turning the wheels would create no friction. The car would go straight. On the road, friction keeps the wheels from sliding. It pushes the car away from a straight line, making it go in a circle.
