Thanks for the answer, Noam. I tried it, but no luck.
I started from a cold boot and opened, resized and closed IE per your instructions. Two iterations of iexplore.exe showed up in Task Manager, but closed when IE was closed. AHP opened two (new) iterations when I used the "Activity Monitor" function (which seems to work as it should) and leaft them open when the monitor window was closed. Starting clean again (reboot, checked for open iexplore.exe, reran AHP), both "Find Other Computers" and "Hardware Configuration" still produce the one-pixel-wide window. Neither one opens an iexplore.exe.
Another thing I noticed during this set of tests. The narrow windows show a blue top (large) and bottom (small) area, sized just like the top and bottom blue margins of a typical windows window. They respond to a click on the window with a change from light blue to darker blue, as if the "window" is changing color in response to getting and losing focus. The narrow window also displays another AHP iteration on the taskbar.
Any other ideas?