The way it generally works: If your email is set to push, you'll hear an audio ping when email arrives, even in sleep mode (plugging in to charge not required). If wifi were asleep, how would you be getting any email, and then an alert? People have even complained because their iPads continued to ping while they had their iPads in sleep mode, because they didn't know how to turn off the audio pings. That's why Apple created the Do Not Disturb feature on iOS 6.
FWIW, what eventually worked to fix my issue was a force close of all Apps. Hope this helps.