To answer your first question this is called a 'push notification'. Here is one resource I found on how to enable them http://www.solvemix.com/index.php/android-smartphones/516-samsung-galaxy-s6-gmail-push-notifications-are-not-received-solution
As to your phone not connecting first: I do not belive this is possible due to the fact that by definition your phone is mobile which means that it's IP address will change constantly as you cross cell sites or connect to different Wi-Fi hotspots. At a minimum your phone must tell the server where you are when your IP changes. These messages are very small (less than 100 bytes) though and should not affect your bandwidth much ( you would have to send 1,048,576 of these messages to reach even one megabyte ).