One, I can concede Janeway just used the comment 13 including Tuvok's for convenience's sake. She wasn't in her right mind at the time so details weren't high up on her list.
Two, I can accept them reporting to Tuvok because let's face it... Chakotay is just the token Maquis to keep the rest in line. A former terrorist is questionable, but she knows Tuvok is loyal. If Chakotay was handling that stuff, well he might start getting ideas of being Janeway's equal.
Three... well let's see. Helm, Operations, Security, Engineering, Science, Medical. That's six right there. Maybe Neelix is his own "department" with food resources. Maybe science is divided into separate departments like astrology, biology and what not. Maybe security and tactical are separate. There has to be a shuttle and torpedo building department for all the replacements they went through. So yeah, I can buy the number being thirteen.