Page Manager and Panels settings — where do they live?
This ended up requiring API calls, since Panels and Page Manager definitions can live in code or in the database. 

First I got the tasks with page_manager_get_tasks(). 

This gave me a list of tasks, which appear to be high-level Page Manager stuff. My Workbench, Node add/edit form, Node template, User template, User edit form, and Custom Page were the tasks.

Custom Page had subtasks that included all my custom pages, so I did page_manager_get_task_subtasks() on each tasks that had subtasks, and then page_manager_load_task_handlers() on each subtask. Within the handlers, there is information about the Panels and other objects that go on that page.