I see what you mean about working with that template... it's not just customized, it looks like they threw away the standard main template files and wrote totally new ones. Nearly all of the elements have different names; some of them may be built with (mostly) standard code, but I can't be sure from this end.
Is the left column turned on in Configuration > Layout Settings?
If getting sideboxes to appear via Tools > Layout Boxes Controller doesn't work, you might be able to add your list to whatever file the current sidebar is built in, with a test wrapped around it so it will only be processed on the one page.
PHP Code:
if ($current_page_base == 'index' and $_GET['cPath'] == 5) {
//your links
}
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