2. It shouldn't damage transmission. Be aware of one thing those it's not always best to coast in neutral. Many newer cars kill the injectors when the vehicle is a. In gear b. no foot on accelerator c. Roma over a set point (1000 Rpms in my jetta). So basically the engine down shifts the cars inertia keeps the motor turning but there is no fuel being used. Sometimes you can coast in neutral if your coming to a stop and it's far. If you realize your stopping sooner than expected or going down a hill instead of using your breaks in neutral keep it in drive and let the engine break it.
Partly right. shifting into neutral WILL cause you to need to replace transmission linkages, etc sooner on an automatic, as they are not designed to be manually switched. That repair, depending on the vehicle, is FAR more than what you will save in fuel over the life of your car.
TL;DR: DON'T MANUALLY SHIFT AN AUTOMATIC!