I am not very familiar with Wordpress terminology yet, so this may be why my Google results come back empty but I guess this is a question that troubles a lot of developers with Drupal background that move to Wordpress.
The short version:
"Is there a way to programmatically add a menu item like mywordpresssite.com/my/custom/url/%/% (where % are wildcards) and define what this page will return in my code?"
The extended version:
In Drupal there is a hook called hook_menu where you can define a menu path like the one mentioned above and a callback function that will take the wildcards as arguments.
What I'm trying to do here is simple: create a button on every post of the site; when this button is clicked in a specific post, it will do a simple GET request via AJAX to a menu item like: mywordpresssite.com/ajax-calls/%current_post_id which will allow me to do some further processing of the specific post in the background.
I hope that the above makes sense. If this is not the way that things work in Wordpress, please provide any resources as to how I should go about approaching the task at hand.