If someone becomes a section moderator, does he automatically get mod powers on all sections of the site? Does he get administrative powers?
Every moderator that's into the group "Staff" can moderate all forums that are moderated by that group, which excludes only a few archive forums. So yeah, you can tell every staff member has somehow full forum powers. If there's the name of a specific moderator under a specific forum, that means they focus on there, but they have powers over all forums.
However, not all moderators can moderate all sections; the list of who moderates what section is located here, along with the forum focusing list.
Also, moderators are restricted from many admin powers, such as permabanning, name nuking, password resetting etc. And those actually do have powers over all sections and archived forums.