OK. Problem solved.
The problem was a .htaccess file that somehow was blocking the operation of any drop down menu. That included category drop down menus and the administration drop down menu (especially when I clicked on "Dashboard") When I clicked on dashboard, it didn't do to the dashboard, instead it just stayed on the home page. Category drop down menus didn't work, but category list menus did!
So, Ultimately, JD was right. It was an environmental problem. A .htaccess file in the root of the domain was preventing proper action. Once removed, or edited, everything behaved normally.
JD sent this explanation:
As I understand it, htaccess files are cumulative. lets say you have
root
*
*********
* *
folder1 folder2
*
********
* *
folder3 folder4
and they all have an .htaccess file. anything running from root, will only have
root_htaccess
applied, but something running from folder1 will have
root_htaccess + folder1_htaccess
run, while something from folder4 will have
root_htaccess + folder2_htaccess + folder4_htaccess
applied to it. So you can see where issues can come up.
If you have a wordpress install for one domain in the root and one in folder1 and they both have permalinks set, weird things can happen.
Hope that explains all. The main thing is it was the .htaccess file that was messing everything up. So with that in mind. Happy building.
Oh. And do donate. The kind of help you get from Larry, JD, and Flynn is invaluable! And really, Atahualpa is the best theme out there.