1) They are piezoelectric sensors, which incorporates a kind of crystal that responds to pressure and vibration, turning string vibrations into an electric signal that can be amplified. They are more commonly used in acoustic guitar pickups.
3) Yes. GraphTech has shown replacement graphite saddles for Floyd Rose trems that have embedded piezo sensors. They should be available to the public soon.
4) They give a more true representation of what a vibrating string sounds like, compared to magnetic pickups.