Is it possible to add more levels in
Content -> Levels to show in the OOTB Table of Contents web part? Could this be edited in some file or do I need to build a custom web part in order to get more levels displayed?
You could try to export the webpart, edit the property 'levelToShow (
<property name="LevelsToShow" type="int">10</property>) and import the webpart again.
You probably want to update the XSL-stylesheet as well, to accomadate for the extra levels.
(Don't change the settings with 'edit webpart' afterwards, you will overwrite the levelsToShow value.)