Can my iPhone automatically send me an email when I turn it on?
No, the phone can't email itself. He probably went into your accounts, checked your email address, and made a backup of your notes by emailing the stuff to your email address. That's what people do sometimes to remind themselves about stuff the next day or later when you read your emails. 

If he sent it to your email address, and then read his own email, it is flagged as read/seen, and won't appear as a new/unseen email on any other email client. So this means you could have missed the notification, but he sure did email it to your email address from your email from your phone. 

Hope this explains it. :)