To 'pull off the road' means to leave the road - your first meaning. So - you can pull off the road onto the shoulder - e.g. to drop off a hitchhiker. You can pull off the road into a carpark. You would not usually use it to mean turning into another road. It's a pretty common phrase - no context is needed! The other meaning relates to objects - e.g. you can pull the lid off something. You can also pull a dog off another dog! You would only use it in this way if the car was broken down, AND you were pulling it (rather than pushing it) e.g. by towing it with another car. But that would not work in your example, as it does not fit the context.

Why is that car pulling off the road. This means that the car is leaving the road (usually leaving the asphalt and driving on the dirt beside the road). This would be said if you're driving in your car and you see a car in front of you suddenly getting of the road. Why is that car being pulled off the road. This means someone is stopping the car, probably a police officer.

Without context I would say "Why is that car pulling off the road" means that the car is turning off onto a different road. However, if it says "What is that car pulling off the road" then something is being dragged off. Hope this helps!