I drive a high milage automatic, it is electroncially shifted, so it isn't like the older style autos. Anyway, bought the car with 225k, currently have 260k, been neutral coasting it quite a bit, and quite often the engine is off. Not really sure how much damage that is causing, since typically the auto transmission will go before the engine in the Toyotas. I'm pretty sure the manual says to not to flat tow as well (does for the 92-96 camry, very similar car, don't have a manual for the corolla).
From my standpoint, if you can put your transmission in neutral, and re-engauge gear smoothly, it shouldn't be stressing the transmission much, engine off vs on is your choise, my car was only $650, I can afford to replace it if needed, and transmissions are around $100-200 in my area, I can do the work myself.
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