Mine does calculate calorie burn but you would have to look on fitbit's website to get more information about how they measure (it measures BMR, so it adds all fitness calories burned to whatever calories are burned at rest.)
It doesn't ever turn off, but does go to sleep. I think all new models have auto sleep, so when you don't move for an hour or so it automatically goes to sleep mode and starts tracking your sleep. As long as it's not charging and you are wearing it, it tracks everything. Even though it doesn't turn off if it's not worn, I think looking at the display eats the battery most, so it shouldn't be burning too much charge.