Question

When the car is skidding, does kinetic friction act on the tire?

When the car is not skidding(wheels rolling), does static friction act on the tire?

If so, explain why.

Answer 1

yes kinetic friction acts on the tyre

the surface of the tire (at the point where that surface meets the road) is not, in general, sliding along the road.
so, in every little surface area element of this kind, we're talking about static friction.

indeed, the tire is rotating because the car is moving forward, but think of this instead as little surface elements of the tire meeting little surface elements of the road. there is no sliding there in general.

if the car is skidding, or "pealing out", or "drifting", then of course kinetic friction most definitely comes into play. here, the little surface elements of the tire are not meeting single little surface elements of the road. the little surface elements of the tire are being dragged across a large number of little surface elements of the road. -> kinetic friction...

of course, turning your front wheels in order to steer is kind-of an in between situation

2.yes there is static friction

As a wheel turns successive patches of the tyre are being placed down onto the road surface and then lifted off again .. normally there is no relative motion between the patch of tyre and the road surface.

When there is relative motion, and the tyre is scrapped across the road, there is (usually) a screeching sound and smoke due to the heat generated .. this applies in straight line acceleration with wheel-spin and when sliding during excessive cornering.

It is the static friction that resists the sliding of the tyre across the road surface .. so it is the limiting force for the centripetal acceleration required for circular motion.

And so it is with braking .. as soon as the static friction between the patch of tyre placed down and the road surface is exceeded