I changed the door handle on the driver's side of the car!
After taking a hair dryer to the door, it mysteriously came open. Turns out it wasn't truly frozen after all, but something inside, in the mechanism, is bent. This is on the passenger side, but the driver's side door has been broken for almost 2 years and we've just been dealing with it because the passenger side always worked. The passenger side is the one that we thought was frozen shut, but as it turns out wasn't frozen but bent. This is bad, since if we can't get in at all we can't use the car.
After the door opened (and it only opened the one time, the first time I managed it), I opened up the driver's side door and we went about the process of jump-starting the battery. Jon was fiddling with the door, trying to figure out what the problem is and why it wasn't working. It seemed to be working just fine, so why it would no longer open from the outside was a mystery. Even looking inside the mechanism didn't help much, since moving the handle made the mechanism move.
Once the battery was charged and the car was running, we went to a parts shop to see how much a new door handle would be. It was more than we wanted to spend ($40) but worth it if it means that the door will work. Then we came home, and I took the old handle off and put the new one on! It was such very cold work, by the time I was done I couldn't feel my fingers or my legs. Jon helped some, but that is one of those types of jobs that a second person is more in the way than helpful most of the time, so I mostly did it alone. But I wanted to, I had said I could do it (my exact words were, "how hard can it be?") and I wanted to do it.
And I did it!! The driver's side door opens like a dream now, it's never been this easy to open, even before it was broken.
Also, opening up the driver's side door allowed us to see that one of the rods in the door is supposed to be straight, and is bent on the passenger side. Now that we know what the problem is, we are going to go easy on the door and not even try to open it from the outside. Especially now that the driver's door works so well. I'm so excited that I fixed it, even if it did make me so darn cold.