Thank you for the proper description, I will try to help as much as I can.
1, 3. – From the screenshot I see actual taxonomy terms as results, but I believe you don't want that. I guess you only want to search the taxonomy terms related to the post, but only return the term objects. For that, make sure that nothing is enabled on the General Options -> Behavior panel.
Also make sure that the “Search in terms” option is enabled on the General Options -> Sources panel: https://i.imgur.com/A4AAjvg.png
2. The results page layout is fully controlled by the theme you are using (usually the search.php file in the theme directory). So for this you will have to make customizations to that template file.Best,
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