Ok first, you can't use index.html if you want to use dynamic features of X-Cart. An .html page is a static page. You shouldn't be using a redirect, you should have this in your .htaccess file:
DirectoryIndex home.php
That won't screw up any of your scripting.
Next, you don't want to use a hits counter that people can see, very 90s
If you want to install tracking software, live chat or a clock, the scripts will probably go in customer/home.tpl, as that is where the <head> tag lives, and most JavaScripts go in the <head> tag. YOu will probably have to surround them with {literal} tags, like so:
{literal}script here{/literal}
Finally, having multiple JavaScripts on your page can cause compatibility issues, as some scripts don't work well with others. LiveChat sounds like your most important one, so get that working first and then worry about the clock and counter.