SSL Certificate and IP address

Currently I have my site running using an IP address. I haven't moved my domain to the site yet. I've had my ISP setup the certificate and that seems to be working. But when I edit my config.php files the site goes crazy. It says my files are not secured and won't show the images and style sheet data. My question is do I have to run the site using the domain name before the certificate will work... Obviously the cert domain name doesn't match the IP address it's setup to look for the domain name. Also do I have to have to move the site to the httpsdocs directory as opposed to the httpdocs directory? I'd hate to have to maintain two separate directories.

Thank you for any assistance.

My question is do I have to run the site using the domain name before the certificate will work... Obviously the cert domain name doesn't match the IP address it's setup to look for the domain name.
Yes befor you can check this without a warning about the mis-match you will have to have it at your domain name

Also do I have to have to move the site to the httpsdocs directory as opposed to the httpdocs directory? I'd hate to have to maintain two separate directories.
A better question for your Host - by default Zen Cart does not require this - But there are many server configurations that might have differing requirements

Thank You!