What path does an asteroid take?

The asteroid will always take the path minimizing its Lagrangian, known as the least action principle in Lagrangian mechanics. With conservative forces, there are four basic shapes for two different cases. Either the asteroid is trapped in the gravitational pull of an object, or it is not trapped. If the asteroid's energy is low enough to be trapped, then it can have either a circular or elliptical orbit. If it is not trapped, then this 'free asteroid' (much like a free electron) can have either an parabolic or hyperbolic path. Much of this information came from "Classical Mechanics" by kibble and Berkshire (although there are much better books).

Depends on the asteroid. Most of them in our solar system rotate around the sun like we do.

The path, as with all the planets is an ellipse.