I'm doubting the second part is true for iPads. I certainly get push notification on my Wifi only when it is sleeping. When I had my 3G iPad 1, it got push notifications when sleeping, even if the 3G was turned off.
I'm not certain about how iPhones work, but since they have to have the 3G on all the time and have a smaller battery, it would make sense to shut down the WiFi under certain conditions. It does not make as much sense to do this with the iPad. You are better off shutting down the 3G, which tends to uses more battery than the WiFi radio, especially if your signal is weak.