if she's drivin your ass around, you don't "have" to open the door for her. But if you are an extra type of gentleman/scholar. You will find a casually awesome way to open that door and get those points.
Take your cue from her. If she sits there waiting for you to open her door, then you should. If she opens the door herself before you have chance to, she doesn't want you to open her door for her.