Well, let me hold off on PMing anything for just a bit. Perhaps they have been making changes at the hosting provider, because now the issue has disappeared.
At their request, we had manually changed the value for $sef from 1 to 0 in the configuration.php file to test. We had to do it manually by editing the file because we could not log into the Joomla admin area. For the record, $sef=1 is the default setting made by Joomla during installation.
With it set to 0, it worked. Now, when we changed it back to 1, it still works. However, there are a couple of new issues that we noticed, so we don't know if those are related or new issues. In any case, we may not be able to continue working this way very long, as we need to move on with our development of the site, and this is starting to consume too much time.
I like to help resolve issues when I can, that is just pretty much in my nature. That said, time is money, and time I spend resolving this is time I am not making money, and there are plenty of other hosts out there that are not having this issue. I am not sure if any use LiteSpeed; that is not my primary concern. My primary concern is that I am doing way too much thinking about something I should not have to expend one brain cell considering.
I appreciate your expected understanding. Again, I will try to help you if I can, although that is not my primary concern.