OP, I have no idea if shifting in an auto will actually damage anything. Go ask your car manufacturer?
What I know is that several years back when I was driving with a company car I always shifted to neutral, or went into lower gear on a downhill (those 1-2-3 numbers on autos), and that car was an auto. Now I didn't work long enough to know if that did any damage, but the car felt fine the entire time. I used to always shift in neutral or lower gears because my personal car is a manual and it's a habit you hardly get rid of, unless you stop driving a manual.
In terms of fuel efficiency, don't know about autos, but with my manual it's simple. If I have to stop and I know that the current gear I'm in will be enough to reach (and most likely I will also need to brake), I leave it in gear and let the engine break do it's job. The engine consumes 0 fuel (compressions keep the engine going), and I save a lot mullah on brakes (I still have my original brakes, at 50% wear after 8 years and 80k kms). If I have to stop and I know that I would need to use the accelerator to reach the stop point because I am too far back, I pop it into neutral, since neutral will burn less fuel for the distance, vs using the accelerator longer just to only use engine brake for a shorter distance.