Hi,
Thank you for all the details!
For this you have multiple options:
1. To create multiple search shortcodes, whith different configurations. You can configure each one as you like (select only the desired post types, exclude anything you need etc..). This is very easy and you can customize each search as you like.
2. To use a single search, and change the query parameters programmatically depending on the page parameters. This is much more difficult, requires programming knowledge.
On our ends the plugin works perfectly on firefox as well. Make sure to update your browser, clear the browser cache and check any browser extensions, which may affect the page – that should resolve the problem.