"When "typing initiate filter/search" is set and ATH sidebar is opened, the search box isn't automatically focused like it should be"

One needs to focus the sidebar (by clicking on it) in order for the sidebar to capture keystrokes. This is an implementation detail of Firefox, and needs to be this way else you wouldn't be able to type into a form on a webpage for example.

Focus is either on the webpage or the sidebar, can't have both.