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#1, Post grid search use default parameter “s” and by default, not that advance search result, there is a filter hook for query arguments, you could take advantage to make advance search form and search thought Post Grid, please see the documentation
#2, not for thumbnail, you can add custom class to item wrapper by filter hook, please see the documentation
There is also HTML element under layout editor, you can pass any shortcode to generate custom HTML each item(post) of post grid.
Let me know if you understand correctly.