No, they definitely have batteries in them. When you touch the capacitive sensor in the car, the car generates a RF pulse which pings the fob. The fob detects the ping from the car and then it transmits a signal comparable to you having pressed one of the buttons on an old style fob back to the car. With a 2-way range extending radio link you could definitely make the key talk to the car over a much longer distance.
Yep. low frequency radio waves. The system just monitors the strength of the signal to determine proximity and the correct passkey. Some cars like Nissan do use RFID as a backup in case the battery dies. There's a hidden key in the fob to open the door and you place the fob over the button to start.