Hi wilson1973,
The issue is when Joomla goes offline, it causes the whole framework to go offline, so you may not be able to get the livechat window to work the way you want when your website is marked as offline through the admin. The only way to accomplish what you are trying to do is to use the hosted version at UltimateLiveChat.com.
As for customizing the hosted version popup window; you can customize the entire look and feel of the popup window via CSS. The only thing you can't do with the hosted version is acccess the PHP files on the backend, but that should only be needed if you are planning on doing some major customizations... for changing images and colors, you can accomplish that by overriding the CSS. You can customize the CSS by logging into your UltimateLiveChat.com account and then clicking on the "Manage" link next to your livechat instance, then proceeding to the Settings page where there is a box at the bottom to override any CSS properties:
https://www.ultimatelivechat.com/my-account.html