I think I might have just solved it. I'm not certain yet, so if you have
any ideas, don't hold back ;-)
For over a decade, I have always used Focus Follows Mouse, and it is
definitely very nice, when numerous application windows are open at once.
You just move the mouse over to an exposed corner of an underlying window
and presto! it moves to the front.
However, when a window generates a password entry window or a message
window or something like that which must always stay on top until it is
responded to, either by entering the password or cancelling it, then Focus
Follows Mouse is not practical.
Firefox, however, seems not to honour Focus Follows Mouse with respect to
the password window, but multiple opened firefox windows do follow the
Focus Follows Mouse setting, so firefox is able to correctly discern the
difference between notification/input windows and the application windows.
I have also played with the Focus Stealing Prevention setting, but I don't
know if this might or might not help me get the effect I desire.
So, to pose the question again: Is there some combination of these or other
settings that will get the desired behaviour?