There is gmail push notification. The notification appears in your notification screen and a @ symbol appears on your lock screen. There is also a sound.

Wi-Fi is always on unless you choose power saving mode. You can also configure it to use 3G automatically when you are out of wi-fi range or if you prefer, manually. Wi-fi will auto-connect when you return.

There are some issues with weaker wi-fi reception than other phones but it will only bother you in areas of marginal wi-fi reception. This is supposed to be fixed in PR_1.1

Battery life is better than many Android phones. It's not unusually good or unusually bad, just normal with regards to smartphones.