Hi,
1. That looks like some form of a bug to me, but I can’t replicate it on our local servers. Can you please add temporary (s)FTP access? I would like to debug this directly.
1.2 – Well, you need the $_GET query variable for that. That information is sent via the URL. For example if I select the “Casa signola”, then it is sent like this: https://i.imgur.com/eTJakc0.png
That value can be accessed via the $_GET['aspf']['tipologia__1'] variable.
2. The two search bars are considered independent from one another. The only thing that is shared is the search phrase on the results page, because that is a global query variable in WordPress.