I'm not certain, but I believe it is a conflict between:
1. the .htaccess file in the root-level directory generated by WordPress, and
2. the .htaccess file in the "protected" directory generated by AgoraCart.
WordPress generates mod_rewrite lines in the root-level .htaccess file that create attractive permalinks for the WordPress site. When cPanel installs AgoraCart it immediately installs an .htaccess file which for some reason causes the WordPress .htaccess file to throw out a 404 error page if you try to access manager.cgi.
When an AgoraCart is installed "by hand" (a web install rather than a cPanel install), it doesn't seem to generate a .htaccess file automatically... the user does it after logging in. So at that moment all works fine after a web install: the user can access manager.cgi with no problems whatsoever.
But after logging in to the admin panel the user is prompted to password protect the directory. When the user clicks that link to generate a password protection with .htaccess and then tries to navigate to another page within the manager panel, it disappears and nothing but 404 error pages pop up. The user is effectively kicked out of the manager panel.
If we deleted the root-level .htaccess file we got access to manager.cgi, but it broke my WordPress permalinks.
If we deleted the root-level .htaccess file, the WordPress permalinks worked again but then we lost access to manager.cgi.
If we deleted the .htaccess file in my "protected" directory, we regained access to manager.cgi and kept the WordPress permalinks, but lost the "protected" password protection.
One webhost tech figured it out and got it working but I don't know how. I asked him what he did so I could pass it on here, but a different tech takes up the support ticket each time and the person who got my question thought it was a request for help, changed the .htaccess file and now EVERYTHING IS BROKEN AGAIN.
BLAST.
I can work with it by renaming the .htaccess file every time I want to login to the admin panel, but what a pain that is, and I'm not sure how this will affect actual use of the cart.
Sorry this is so long-winded, but I just wanted to tell you want is going on and to give A HUMONGOUS THANK YOU to the little birdy from this forum who tried to help me on the side.