The Earth as a rest frame is arbitrary. So saying that two cars have velocities toward each other is just a choice of reference frame. You are right that you can consider yourself to be at rest while the car coming toward you has some speed which is twice what the Earth frame would say it has.
As it comes toward you, you observe it to have a contracted length. Also, the Earth which seems to be coming at you will have a contracted length. The closer the other car gets to the speed of light in your frame, the shorter it becomes as it heads toward you. This will also happen to the Earth. You may be surprised at how quickly a near light speed Earth travels beneath your car. It not only passes by quickly because of its speed but also because it seems contracted to you, in fact the entire solar system will seem contracted.
This is one of the lesser known aspects of special relativity. If you travel to a nearby star at close to the speed of light, then you experience less time on the trip there. But how can this be? On the ship you feel like time is passing normally. And it is for you, but the universe ahead of you is contracted to be shorter. So you can travel to distant locations in a very short period of time from our point of view. In fact, at near light speed, the distant stars can be less than a light year away. At tremendously close to light speed the entire universe becomes accessible within a few years of time on the ship.
