Thanks for sending the theme. I was able to see the issue for myself locally. There's a fairly easy workaround to this but it might involve some other adjustments to get things working right. If you change the Events Template setting to Default Events Template in Events > Settings > Template, this will get it so the category pages at least show up.
Now, you might need/want to do some customizations to this template to get it to jive better with the theme. To do this all you need to do is modify /wp-content/plugins/the-events-calendar/views/ecp-page-template.php - make a copy and place in an 'events' folder in your theme. Then include any theme specific markup or code from your theme's page.php or other page template you'd like to use, wrapping it around the plugin specific code in ecp-page-template.php
I hope that helps!