Greetings Mohammed
I will answer the questions in reverse order (and I do hope I understand you correctly, so if I do not, please come back for clarification).
You want the results to show in the area under the search, on the same page? This is POSSIBLE in different ways. The search you show appears to be on the home page, yes?
I will explain. S&F Pro has many different display methods, and the one I suspect that you will want to use is the ‘shortcode’ method, as described here:
http://www.designsandcode.com/documentation/search-filter-pro/search-results/using-a-shortcode/
In your case, if it is to be the home page, the page that you create would be the one used by WordPress (in Settings -> Reading) as the Home Page.
In turn, this depends on the theme that you are using being able to accept the shortcodes being entered, or you may need to edit the theme’s template php file to add them. So, this can vary from very easy to quite hard, depending on the theme.
As to the first question, S&F Pro has 2 functions, Auto Count and Hide Empty Values, that mean that as you select values from one dropdown, only relevant values remain in the other dropdowns (the other dropdowns re-populate the values so that the other non-valid values disappear).
This works best (quicker and more smoothly) with Ajax enabled, assuming your theme supports Ajax, otherwise a page reload happens.
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