Our plugin requires the search strings as a query in the URL to work. You can hide them if you set the Make searches bookmarkable setting OFF (in the form Display results settings tag).

V3 of our plugin (due in a few months) should offer an alternative method which may better suit your needs (I think sending the search parameters using Transients).

I think that it might depend on the Display Results method and whether the page is using Ajax. If the page first re-directs to the results page, as your URL would have, then the query string must be there.

I think, for now, it will need to stay that way for your site.

There is not a solution to the issue of categories needing different filters. For category pages, it would be normal to use the Post Type Archives display results method, which allows only one form. I cannot see a way around the need to remove certain filters for each different category, other than custom CSS based on the category class name that appears in the page body tag and use this to hide form filters/fields.

As to noindex, see this post:

https://support.searchandfilter.com/forums/topic/no-results-message-but-results-shown-below/#post-245858