When I'm working on my Macbook Air, I would like to get mail "pushed" to me without manually opening the Mail app. What is the point of the notifications center when the Mail app does not push emails? Is their any way the Mail App in OS X Mountain Lion can "push" emails without manually opening the app? All my accounts are Yahoo that are signed into the Mail App, if that helps.
Probably not. Why does it matter whether the app is running or not? Just have it launch automatically at Login, and hide it, if you like.
Have you checked to see if there's a Yahoo Mail notification add-on or extension for your web browser?
No, but this is coming in OS X Mavericks. Mavericks will display Mail notifications without the Mail client running.