Heh. I think I figured it out. You know how you can make the Application folder open up when you select an application in the dock? You press the command key while you're selecting it. Whoa. I believe my ancient crumb-catcher keyboard had the command key engaged (though the key itself wasn't stuck down). I exercised the key a bit, and things improved. Though it still happened occasionally and, you know what? When I disconnect my keyboard, the problem goes away entirely! I need a new keyboard!!!
I should say that other strange things were happening in other applications, and even when I booted to a clone, the same problems were happening.
By the way, a good strategy for checking out a flaky keyboard is to unplug your keyboard and screen share with another machine. When I did that, using the SAME computer and the SAME applications, everything worked great.