When you click on FileAdd/Remove Snap-ins do you get a long list of
available snap-ins?

It's empty

Are you able to select Certificates and hit the Add button to add to
the right-side panel?

I don't show a right side panel on mine (XP SP2) - there's a dropbox,
a (currently all white, or empty) pane beneath that and buttons
Add,Remove,and About; and OK and Cancel. I can click add and on the
popup select Certificates, hit add there, and "Certificates-Current
User" appears. That's as far as I get, and the window where you'd
expect to see them is empty. And even that choice doesn't persist if I
close/reopen MMC.

Thanks for your interest and effort to explore this.