Same here. It isn't as blaring as Bella, Jacob, or Edward. I would say Ez-mee thought.
I don't know... I'd sooner associate Esme with Twilight than Jacob or Edward. A lot of people probably hadn't heard of it before Twilight, whereas everyone has heard of both Edward and Jacob and there are many other associations with both those names.
As for pronunciation, I would definitely say ez-may. It seems that most people would pronounce it this way as well, so you would have to be okay with correcting a lot of people, a lot of the time.