How do I identify the process that owns a spurious dialog box?

Sysinternals Process Explorer has this feature in the form of a crosshair that you drag over any window or widget. Click the "crosshair button" and drag to the window, then release.

In task manager, you should be able to right click it and select Go to process, which will open the processes tab and highlight the executable:

If that isn't helpful, Winspector (basically Spy++) can give you a better idea:

Check out Winspector. Install it & run. Switch the Window list's tab to Process. Then drag and drop the red "foresight" from the upper left corner of Winspector onto that dialog box. Winspector will highlight which process owns the message box.