When are arrow keys acceptable as navigation tools? Is it acceptable to require arrow keys to use something like a complex menu? Or should they be reserved for performing additional functionality or as an extra way to move around?

Arrow keys are usually expected for arrowing up and down through list options in a combo box, not necessarily a menu. Another example is arrowing back and forth through the tab elements once a tab list receives focus. In menus, you would want to be able to use the tab and not arrow buttons, so more just components where there's a primary structure, like a tab list containing two individual interactive components, versus a menu where it may be linking out to separate pages.