In your case, you have a parent -> child category structure, but all of these terms are within the 'category' taxononmy. A taxonomy is basically a group of terms (like post categories, post tags, product categories).
Taxonomies are a nice way of categorizing things, usually when every item belongs to at least one of the taxonomies. For example a WooCommerce product usually has at least one product category and a product tag – but it is nicer to have them separated.
I usually define custom taxonomies whenever I feel like that the parent category seems redundant. There are no rules on when or where to use them, it is up to the developer/editor to decide.
There are multiple ways to place the results below that area:
I see you currently use the two-column shortcode. Depending on how you want the layout, you could try something from the shortcode generator instead, like for example this one will place the search bar and the settings to the top section, taking 50/50 of the space, and then the results below:
[wd_asp elements='search,settings,results' ratio='50%,50%,100%' id=2]
This will result in the following layout: https://i.imgur.com/w86BycQ.pngBest,
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