I checked your site, and the problem is some of the old CSS for MapPress was copied into the theme’s “styles.css” file. Remove that CSS, or bring it up to date with the newer CSS classes in the “mappress.css” file for the plugin, and it should work.
Specifically, the class “.mapp-ib” is incorrect for the new version. It has a “display:none” setting which prevents the infobox from displaying properly – that’s why you’re not seeing it on click, and it’s also why it only occurs on the frontend.
To answer the question about the templates: The default popup templates are visible in the popup template editor on the MapPress settings screen. The documentation describes how to modify them.