Re: GMail is not push E-mail?
Gmail does use push, at least on Android, but not traditional IMAP push. On Android the push messages are sent through Google Play Services - which maintains a single active connection for all Google services (Hangouts, Gmail, Play Store updates, etc.). GCM messages, which many applications use for push messaging are also tunneled through this single Play Services connection.
The idea behind the single connection is that it helps save battery, 1 connection waking the device rather than a connection per-app waking the device.
Whether you can get Gmail to use push outside of an Android device I can't answer, I only use it for Play Store access really. I know about the above as I'm a developer and it's come up in several documents and/or been mentioned during Google IO sessions.