Soon after I wrote the above, mozilla stopped working altogether.
What happened was I thought "I'll create a new profile, to see if that works", so I did that, but next time I opened mozilla, the "Choose profile" box came up, but clicking "Start mozilla" after choosing a profile resulted in nothing happening, whichever profile I had chosen.
So I made a copy of my user profile folder from .moziila, then uninstalled and reinstalled mozilla. It's working OK again now, all the problems I mentioned above resolved (and some I didn't mention, like import/export bookmarks not working) Rather to my surprise the new installation had all my bookmarks, passwords, cookies, etc, without me having to copy the profile folder back. I had thought that a new profile would have been created with a reinstall, but it seems not.
Anyway - it seems I found the answer without really knowing the question. Does anybody have any theories why this happened? Has it happened to anyone else? Will it happen again? Who knows.
It was Mozilla 1.3, my distro is Mandrake 9.1