Well, I seem to have solved the problem, but it wasn't (and still isn't) obvious.
I reverted to 7.10 and it seemed okay (I think ...), but the Java installation in the middle of the Opera installation made me realize something had been missing. Sure enough, the 7.11 package I had been using did
not include Java[/i].
So I uninstalled 7.10, downloaded Opera 7.11
with Java, and installed it. The Java installation said I already had Java 1.41_01 (possibly a slightly different build?) and did I want to
uninstall it. I okayed it, and after the installation confirmed that I had (or still had) Sun Java installed.
No go, Joe.

Still splattered.
Once more, I uninstalled Opera, then Java and its poxy Webstart companion. Deleted the Opera directory (which I'd been doing previously), then found that the Java directories were still instact, as if the uninstall was for nought. :-?
So I creamed the Java directory as well, then installed 7.11 once again - and all was well.
I'm guessing the residual Java files were screwing things up somehow, so it's a tip worth noting.