Display All Products by Category with WooCommerce

With WooCommerce, I want to display all the categories in a store as headings, with all of their products listed below in an unordered list. Is this possible to do? I’ve seen a few things that’ll let me display a list of categories or a list of products for a specific category, but nothing that’ll loop through everything the way I described.

Setting up an API “listening” page

I need to create a PHP script that will receive data from another application via an HTTP post, process that data, and update the WordPress database. This script will run silently (i.e., no HTTP output). What is the best way to set up such a page? I thought about creating a custom template file with the PHP script and assigning that template to a blank WordPress page, but is there a better method I should use?