Yes, of course, that language would be useful. We've had other people who need that clarification, it seems, so it couldn't hurt to add it.
Maybe so, but I think maybe it would be better to say 2AM to 8AM local time, not "every time zone". Still, the way I read it already tells me it's in my local time zone, as it doesn't give a time zone, just 2AM to 8AM. Being more descriptive certainly wouldn't hurt, though.
Good discussion, I also think it would be useful to specify that the hours listed are in your local time zone. I've sent over this suggestion to the appropriate departments. Thanks so much for your valuable feedback!