Why does rotating a tire forward along its axis make it easier to rotate it? Probably anyone who has driven a car has noticed that it's easier to turn the steering wheel when the car is moving forward than when it is stopped (I've even heard it's a significant wear on the steering system to make a habit of turning when stopped).
What's the mechanics behind this?
I know that the first thing to discard is the idea of an infinitesimal contact patch length, but what comes after that? I can understand that half of the contact patch is going to rotate in the rolling direction when turning and moving, but that seems like it would be offset by the other half of the contact patch moving in exactly the opposite direction.
 A: Each tire is in contact with the road over a finite sized area. (In fact you can easily estimate how big the area is - the air pressure in the tires is supporting all the weight of the car, so the weight of the car = (total contact area of all the tires) $\times$ (tire pressure).
If the car is not moving, you have to twist the part of the tire in contact with the road in a circle, working against the friction force between the tire and the road.
However if the car is moving, you are continually adding a different piece of the tire to the contact area as the wheel rotates, and removing the part of the "old" contact area. A tire has some flexibility through its thickness from the inside to the outside, so in the short time that any point is in contact with the road, the rubber can deform enough so the rubber doesn't have the be dragged sideways across the road while you are turning the steering wheel. 
You still get a some "slipping sideways" at the edges of the contact area where the normal force between the tire and the road is smaller, and that is one of the main reasons why the tire tread wears away as you drive the car - even if you don't deliberately spin the wheels, lock the brakes, or pull donuts at every opportunity!
