Don't ever put a transmission in neutral when coasting downhill. Automatic or manual. Doing so is dangerous for several reasons; already mentioned above.
There's also the element of brakes. In the mountains, you want to be using lower gears to maintain speeds down hills. Constant braking down a long slope is a good way to boil your brake fluid and potentially lose all braking. Never mind a lot of added wear and tear on your brake pads.
I am fairly sure this is all in the owners manual.
And Yes, you'll get better gas mileage with it in gear because the injectors can shut down. When idling, youre always using gas.
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