I am recreating my current directory site so that it is easier to find results (listings) through search and navigation.
I am unsure of the best way to set this up, can you please advise?
Please see the attached slides for my requirements.
Now, as I can see with your requirement you want that you have setup like:
=> Category (archive) => Sub-category (archive) => Post
Where you want that Category (archive - for top level terms having no parent) should display the list of sub-categories and when you click on any specific sub-category, it should display the related posts to that sub-category.
This is a whole requirement so I suggest we should move step by step to achieve your goal. Please setup the archive as mentioned and let me know if you need further assistance.
Before doing this, I was wondering if you can explain and point to guidance regarding the API functionality?
Essentially, what I am looking to do is have a searchable database/directory on my website. Using API, I would like third-parties to be able to call on my information (replicating the same searchable database/directory experience) on their third-party websites.
There is no such native feature available - you may have to add the custom code to integrate with such APIs.
Another solution I could think is adding search page on your site and call that page via iframe on another site.
That is why I ask you to check with our existing reference site on our free test platform http://www.discover-wp.com and check feasibility of your requirement.
Can you please explain how to expose the toolset fields and how to display the listings via the WordPress REST api? Is there any guidance I can follow for this?