Well, *something* would have to be running in order for the number of unread messages to be updated; maybe there's some super-leightweight other app that can do that.
Power Nap is the feature that allows apps to update while the computer is asleep; App Nap is what helps prevent apps using GPU power when their windows are hidden or obscured by other windows.
I'm sort of confused by what you're after with not wanting to have Mail open – it's a good citizen among apps. It doesn't use much active memory and, when idle in the background, it uses virtually no CPU power.