Welcome to the forum!
Another automatic transmission efficiency technique: if you have "sport mode" (ie. gated shifter or paddles/buttons), you can often get the transmission to upshift sooner and get into top gear earlier than the default transmission programming. This works even with CVT's - see
"Short-shifting" a Jeep Patriot & Audi A4 CVT (forcing early "upshift") saves gas
As for your question about potential damage while coasting in neutral while idling, it probably depends on the individual vehicle. I spoke to a mechanic at a transmission shop specifically to ask his advice about the same questions you posted and his only warning was to avoid engine-off coasting in N, but he may have been generalizing to
most vehicles. I'm not sure.
There are some transmission designs where it might be an issue. Just as there are some auto transmissions where you can safely coast in N with the engine off, eg:
https://www.etrailer.com/question-142325.html
Best advice is to know the details of the car in question.