Hi!
1. That I believe is only possible with directly changing the result template, you can find more information about results templating here.
Basically, I would first place a custom code there to get the categories. Then, if there are categories, then add a HTML code just above the title, or wherever it is needed. Let me know if you need further help with this. Since this is a customization request, I cannot guarantee my suggestions will work in all cases though.
1.1. On the Frontend search settings -> Advanced options panel change the first option, I believe that is what you are looking for: https://i.imgur.com/LnpgX6J.png
2. Do you mean sub-items for custom field filters? I don’t think this is possible, I’m not sure.