Having trouble getting mega menu to work responsively
No, that module doesn’t claim to provide responsive menus.
Quite the opposite, in fact; its mission is to provide you with the tools you need to build your own styles, which can of course be responsive if you desire:

The module provides classes and ID's that allow for a full-range of styling customization, from more traditional (but multi-level) drop-downs to full width slide out menus.
A couple demo skins are supplied, but these are purely for instructional purpose to help get you started with theming your mega menu. Currently, you should have an understanding of how Drupal overrides module CSS files and how to add CSS to themes, in order to provide your own styles in your particular theme.

So the module just gives you a framework to start from, you need to provide your own CSS.