It took a while, but I finally decided to read the error reports. When I did so, I found that nearly every one of Firefox’s crashes that I experienced was related to a specific extension. I dug around in the registry and the extensions folder, and found that it was none other than the Java plugin.
Living on the edge, I decided to just remove it. I uninstalled Java, then deleted references to the plugin in the registry and filesystem. Voila. Everything works and Firefox hasn’t crashed on me in over a week.