Hello guys, regarding battery usage, here is the full explanation. What push does is that it keeps a live network connection with Apple's server 24/7. So using push will have some minor battery impact but still better than the fetch alternative. Fetch connects every 15 mins and wastes a lot of battery. In summary here is the battery usage from most to least:
1) Fetch - connects to network every X min
2) Exchange + Apple Push Notification - main TWO network connection 24/7
3) Exchange - maintain one network connection 24/7
4) Apple Push Notification (i.e. PushGmail) - maintain one network connection 24/7
5) No email account at all on phone - no network connections ever
So there you have it. This answers all your questions. Notice scenario 2, if you are using Google sync + Apple's push notificaiton, you are keeping two network connections alive and wasting battery. This is a key downside to google sync. Chances are you will have some apps that do push notification (e.g. news, IM, etc.) and if you use google sync, you are creating an extra network connection for no reason. Whereas if you use PushGmail, all your push apps will share the same network connection and hence save battery.
If you are asking if turning on push will drain more battery, then yes, but it shouldn't be significant. Let me know if this was helpful. Thanks!