A2 : Yes. In order to conduct driving lessons on a professional basis, all instructors must be registered with the DSA.
Some schools use 'PDI's (potential driving instructors). These are registered instructors but who have not yet passed the final part of their instructors test. These will display a pink triangular sticker in the car window.
I am a fully qualified instructor and as such, display a green, hexagonal sticker in the car.