My desktop icons have suddenly disappeared, perhaps having to do with
the update two days ago to this package.

My system is up-to-date from Trixie.

I've tried turning the desktop-icons-ng extension off and back on,
logging out and logging back in, and restarting, none of which has
brought back my icons.

Other extensions I have installed are working fine.

I also tried opening the settings for the desktop-icons-ng extension
from inside the extensions app, and nothing happens, i.e., the
settings don't open.

This is very weird. I'm using my computer this morning when all of the
sudden voop! I see an animation occur on my desktop out of the corner of
my eye, and when I shift my eyes to look at the spot where the animation
happened, my desktop icons are back! I didn't do anything to cause them
to reappear, they just reappeared. Then I logged out and logged back in
and they were gone again.

and while I'm waiting for that to happen I get several pop-ups from
GNOME telling me that gjs isn't responding and asking me if I want to
force quit it.

Similarly, when I click on a file on my desktop and hit the delete key,
it takes like 30 seconds for the icon for the file to disappear, though
it appears that the file is actually deleted immediately.