OK, checked the error log and there was nothing that looked remotely relevant in it. Just a few missing favicon.ico messages and that sort of thing. Plus two occurences of "File is not a JPEG file" from the massedit page, which I think was caused by a couple of images that had spaces after their names (fixed now). Also, just to clarify, it wasn't a total blank page it was showing (like the kind you'd get with a fatal error), but a page with the breadcrumb trail, title, footer, etc. but no actual content. Anyway, there don't appear to be any PHP errors happening.
Also, I checked the code of the index page and it is not assigning the password_protected class to the thumbnail for that gallery at all. And I haven't changed the CSS in any way.
Just to verify whether or not it was a theme issue, I switched back the default theme. This did result in the password prompt showing up - but the image thumbnail for the protected gallery still appeared in the main listing, and did not have the password_protected class. I tried a couple of other themes with basically the same result, and when I found one that included the random image block, it still showed images from the protected gallery among the random ones.
So those two problems are not theme-specific - *none* of the themes I tried (including default!) gives protected albums' thumbnail that class, or excluded protected images from the random image block if they had one.
Then, just to see if the problem with the password prompt not appearing was due to anything I did or not, I uploaded an unmodified version of Thinkdreams - and it doesn't show the password prompt either. So apparently that problem is in the Thinkdreams theme itself, not in any of my changes. Which sucks because it's the only theme that looks relatively similar to what the photographer I'm doing this site for wants. :-(
Any ideas how the password prompt could be restored to Thinkdreams? I'm not afraid of editing code...