How to register child taxonomies?

Too much for a comment, so I’ll leave it for an answer:

1. You’re asking a lot of Qs that should actually be different Qs and not dumped into a single one.
2. (Ad 1.) You don’t register “child” or “parent” taxonomies. You add (child-)terms to a hierarchical taxonomy via the WP Back-End UI. You can do it programmatically with wp_set_post_terms();, wp_set_object_terms();, or other functions, but that’s more internal stuff and tough to handle as you’d need to check on every request if it already exists or not (adds DB-queries), so it’s not really recommended as you can’t do it on theme activation as long as WP doesn’t offer a hook there.
3. (Ad 4.) You can query multiple taxonomies at one time as seen on the WP_Query article in Codex. This queries the taxonomy and all assigned posts. You can also query for terms that are inside a hierarchical taxonomy. But there are no “child” taxonomies. Only “child” terms. You can add those child terms to your tax_query arguments. Doing something like if is child of can be done with functions like get_term_children();. If you link to a term archive, you can modify the query (search WPSE) to include child terms as well.
4. (Ad 3.) As stated in other answers & comments to some of your other Qs, you’ll need to hook in stuff like save_post and similar. You’ll find enough answers here on WPSE that show you how. You’re close.
5. (Ad 2.) You’re mixing labels with actual arguments. The parent_item and parent_item_colon arguments are meant to set your custom text and can’t register “child” taxonomies. They’re only used to modify the default text you see in the admin UI.

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