have you chose the option to disable syncing? usually that happens when that isn't switched off

 

it makes it buzz as a notification, but shouldn't make it turn on at all. Wait do you mean the phone was physically turned off as nothing will work unless you turn it on? if so maybe something(maybe a feature?) turns on the device when connected to a charger.

 

does the phone boot when yu connect it to its charger? also turn it off during the day and keep an eye on it if it turns on by itself.

 

ive never heard of a email turning on a phone ever. turning off sync should prevent all this though.

 

For the phone to go from fully shut down to on sounds mega strange tbh

 

sometimes the kernel forces the device to boot once the phone is plugged in, this happened on the Optimus M.

 

Dumb question, but even with Flight Mode turned on, you can still connect to wifi. Are you connected and having email sync?
Are these actually new emails or old ones being sent even if read/archived? Is this your Gmail app or the stock/third party app?

 

Pull down the notifications shade, tap the speaker toggle until it says mute, then you should have no sound OR vibration. But ya, you probably just had your phone in sleep mode and not turned off entirely.