Well, it seems that some adware was actually disabling Internet Explorer.
When it crashed (right after bringing up the stupid pop-up ad), Microsoft
sent me to the Pest Patrol website.
I downloaded their software, and it found 49 examples of adware and spyware
on my computer! I deleted them all, and we'll see if things stabilize. The
pop-up ad that was appearing when I opened my website is gone, at least.

Now, can someone tell me how to disable whatever it is that wants to
load Microsoft Internet Explorer when I click on a URL? I don't want
Internet Explorer on my machines.