I have never understood this part: How is it that Tankannahai managed to mutiny and subdue Barlahan with the help of the entire ship's crew (as far as I can understand) if up to this point no one inside the ship liked him? Even if I had the help of the Net, getting control of the ship without anyone else's help would have been impossible.

Whether or not you actively restrain the captain, or disobey his orders to his face, taking unilateral control of the ship -- even if it saves the ship and all the crew -- is still mutiny. Tank used his position as Chief Engineer to get the crew to do everything he needed to take unilateral control away from the Captain. Most of the crew might not even have been aware of the issue because Tank was setting things up for a last-ditch escape maneuver. They might even have been told that, which would sound perfectly sensible to them. As rigid as the society seems, they wouldn't have thought Tank was committing mutiny until the Captain ... exploded off screen ... when the ship didn't do what he'd commanded. When the Captain exploded, they would have realized that he wasn't in on the setup. Not being suicidal themselves, they might well have put him in restraints without Tank even being aware of it. Of course, that's ANOTHER case of mutiny, but it's entirely separate from Tank's actions.