The recommended way to customise the CSS for the plugin is to copy the content of the stylesheet included in the plugin into your child theme's stylesheet. Then prevent the inbuilt stylesheet from loading by adding the following line into your child theme's functions.php
<?php remove_action('wp_print_styles', 'cc_tabby_css', 30); ?>
Changing the responsive breakpoint for the elements generated by the plugin won't cause any problems with your responsive theme.
The plugin was specifically designed to work with & play nice with responsive themes. The original version I built for using on sites for my own clients' sites didn't actually include a stylesheet as I was putting the CSS directly into the stylesheet. I added the stylesheet for the public release as a "get you started" measure for the plugin's users with the intention that most people would want to use a tweaked version of the styles.
I'm still seeing the minified version of the stylesheet on your site, so its not possible to troubleshoot the tab layout issue yet. I'd guess you need to empty the cache.