ntaylor, I looked over the files the best I could. Apparently the theme works as a whole, but my personal opinion says that the theme code is a mess.
Regarding the original topic at hand, our plugin doesn’t handle the display of Custom Post Type data anywhere. We focus strictly on the registration of the post types and taxonomies. Why they’re not displaying properly in the chosen theme is beyond me but it may be best to talk to the theme developer if you’re using some display method that they coded, or if it’s a plugin you’re trying, talk to the plugin author.
The post type archive and taxonomy archive are both going to use archive.php first. Once you get into a single post, the custom post type would then use single.php There isn’t a single term archive for taxonomies as best I can tell, so it’s still use archive.php until you get to a single post. Categories would use category.php first, then archive.php.
I hope you are able to get it figured out, but I’m 99% certain it’s not because of our plugin itself.