Hi!
1. Do you have any taxonomies? I can’t see any. This function uses the wordpress built in get_taxonomies function, so it should return the taxonomies if you have any registered.
2. Categories doesn’t have images by default. If the image is stored as some kind of custom field, it might be possible to retrieve it and create filtering function based on this article: https://wp-dreams.com/knowledge-base/showing-the-post-type-name-in-post-title/
3. I just tested, it works for me. Can you explain when it’s not working? I tried it several times on your frontpage, and it worked everytime.
4. It was because empty categories are hidden in wordpress by default. I made a quick modification, now all the 136 categories are visible. (even the empty ones)
5. You need to exclude all those categories which you don’t want to use on the “Advanced Options -> Exclude categories” as well.