Thanks so much for your help. I figured I'd give an update. It turns out it wasn't the weatherstripping. We have a Saturn Ion, and evidently there are drains all around the sunroof that drain to the wheel well. These drains clog notoriously easily, and when they do water backs up and the car floods. Which explains why I could figure out where the weather stripping was bad, and also why it only leaked during huge storms, and then very dramatically.
So, they want to charge us a couple hundred dollars to de-clog, but DH is going to try to do it using a plumbing snake tomorrow. I'm really excited to see if it works.
Drying and mold fighting is still a work in progress.
In good news, we completely paid off the car! So, while it's a bit junky now, it's 100% ours.