Theme div.item-list of views pager? How can I theme the div with a class of item-list that appears around a views pager? All I need to do is add an additional class to this div when it surrounds a pager. 

I can find lots of documentation on theming the content of the pager, but none on how to theme them this div.
Typically, unless you are using a specific contributed module, the pager is a template file that comes with views. You can override theme_pager for the specific view (or all your paged views) and add the additional div.

https://api.drupal.org/api/drupal/includes!pager.inc/function/theme_pager/7

Examples on how to do this:
https://www.drupal.org/node/387786