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When you key off to EOC, you need to turn the key back to 'run' so it powers all the other bits of your car like airbags and odometer. This also prevents the steering wheel from locking when turned!
If you're coasting downhill and its in gear, you'll engage DFCO (deceleration fuel cutoff), and you should see 9999 on the scangauge. At this point you aren't using any fuel. This is perfectly fine, and it won't damage your transmission.
If you set your 'fuel type' on the scangauge to 'hybrid' it will stay on with the engine off.
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