Hi,
Thanks for your detailed post, to start, I’m assuming you’re referring to Yoast primary category since WP & WC don’t define a primary cateogry by default.
For the hierarchy part, can you try this shortcode:
[alg_product_terms taxonomy="product_cat" orderby="hierarchy"]
It should get you the categories in hierarchical order, once this is solved, please send me at [email protected] to see how specifying the primary category can be done.
Omar