The taxonomies filter seems not be working. Even if a filter is selected, it keeps showing the same results. If there is no post on a taxonomy, shouldn’t it show “no results”? Also, if one chooses only a taxonomy filter (no keywords), shouldn’t it redirect to a results page with the posts related to that category (or “no results”, if it is the case)?
By default there is an option enabled, which basically allows results that does not have terms from the selected taxonomy for filtering. If you turn off this option, then only results that has the terms from the filter will be allowed: https://i.imgur.com/yZbi3Tv.png
The live search will only trigger when no keyword is entered, if the character count option is set to the default 0. You can change it here: https://i.imgur.com/Lm82oed.png
Right! About the first one, however, it worked with the live search. Why on the results page (“Search for”) there still are results that do not have terms from the select taxonomy? By the way, if a filter is selected and no character is entered, it shows even posts that are not from the post type selected in the admin area.