On my Lumia 920 running Windows Phone 8.1 Preview for Developers, the option to sync Gmail emails "as items arrive" is definitely available:
When I enabled that option (normally it's set to manual sync as I don't really use Gmail) and sent myself a couple of test emails, they showed up almost immediately in the notification centre as well as on the live ...
No, push notifications do not need a specific background task for each app wanting push notifications, for updating live tiles or sending toast notifications (there are developer samples on MSDN that demonstrate this without the background task).
Some applications may use raw notifications, which are only delivered if the app is running. For example, ...
Go to Settings->System->Notifications & Actions.
Make sure Notification for Whatsapp is turned on.
Also enable Show notification in action center for the app.
Adjust Banner notification, sound and vibration setting to your needs.