Okay, so explaining this problem in words is nearly impossible but I'll try. Essentially what happens is I turn on the phone notifications of texts and emails start flashing and then they're gone and they come back and disappear, over and over again. I can't click on these notifications, and I can't close them. It's really weird. Then the phone after a minute or so shuts down and turns back on by itself and the cycle goes on until the phone kills it's own battery. Here's a video: http://www.youtube.com/watch?v=ZRl8YJX8xoA

You could try deleting all messages if the phone is still responsive, but it seems something is severely broken. Backup what you can and reflash

I've already deleted all my messages and emails, as is demonstrated in the video.

If you've got Maenotify installed you need to remove it.
I had a similar problem (although not as bad) when I used that utility. Notifications went round the tank even after clearing missed calls, emails and SMS inboxes completely.
If that isn't the case, then as qwazix says, backup & reflash. But I would avoid restoring apps/app settings in case the problem returns.
Reinstall your apps from fresh over your own Wi-Fi.