Dude, I have a 2006 Corolla with 120K+ miles and I'm going through something very similar (my front decal was stolen too...very strange). No problems until this fall. I was out of town for a while (a few months) and the car sat, and it hasn't started on its own since. I jumped it and let it run, but it kept dying. One of the times I had jumped it, I started driving and it died while I was making a turn. Scared the absolute **** out of me. Was able to get it to the side of the road and then to a dealership. They say I need a new alternator and some other random stuff. Sounds like there was some sort of slow electrical drain while I was out of town.
I'm debating what to do. This was my subway car; I live about 2 miles from the DC metro, so I drive from the house to the parking garage and back. Talking with my friends who work on cars, it sounds like I should be able to get another 120K miles out of it, but I'm also thinking about upgrading my Rav to a Highlander and I could try to use the Corolla as part of the trade in. I have one of those jumper box things, so I may try to charge the car and see what happens.
Good luck!