Hi!
Thank you for all the details, it helps us a lot.
1. & 2. It looks like you have either a misconfigured .htaccess file or there is an incorrect template redirect rule defined within the theme functions (or perhaps another plugin).
If you try to visit the search page with an empty phrase here: http://foodmixtest.net/rdf_portal/?s=
..then you get automatically redirected back to the main page. Unfortunately that is not the plugin doing, you can verify by temporarily disabling it.
3. You can do two things here, one is to increase the minimum character count to trigger the search, as well as change the placeholder text to something like “Enter at least 3 characters here” or something similar.