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.
In order to reach the earth's surface, the asteroid has to first enter the earth's atmosphere. This happens when the asteroids path intersects the earth's orbit around the sun. Second, the asteroid has to reach the earth's surface without eroding. As the asteroid enters the atmosphere, the high velocity of the asteroid increases the drag force, thus causing more resistance and erosion. Generally, asteroids become small before hitting the earth's surface.
Yes a an asteroid can destroy a planet, although it is very unlikely. A large enough asteroid, moving in a path which strikes the planet at right angles, could generate strong enough shock waves to disrupt the planets core, causing it to break up. What is more likely is that the impact will alter the planet's orbit, as well as affecting the environment in detrimental ways.
Well you never know until it happens. Eventually, some large piece of cosmic debris, whether an asteroid, meteor, or comet, will strike the Earth. However, it may not happen for millions of years. Such events are hard to predict. There is also some possibility that the human race will observe the approaching asteroid and send out a mission to alter its path, so that it will miss the Earth.