You need the key to open the door and start the car. You just don't need to _use_ them. The key has to be in the car to start the engine. The key needs to be near when you push the button to open the trunk (not the button on the fob, the button on the trunk). And you need the key nearby in order for the handle to unlock when you grab it, or for the button to lock the door.

And that's how the button works, btw. The door unlocks automatically if you have the key in your pocket and you just reach out to open the door. When your fingers touch the back of the handle it should unlock while you're opening it. And to lock it you can just push the black button after the doors are all closed. This one I didn't quite grasp myself at first, but now that I know these two things, I hardly ever take the key out of my jacket. Just grab the handle to unlock/open, just push the button to open the trunk, and just push the little black button on the handle to lock it up.

Easy as pie.