I have no doubt that even the most republican of cultures launched from Earth would end up thoroughly monarchistic by the time the generation ships arrived at their destination. At best monarchistic - who knows what savage new forms of society could evolve in that sort of context?
More optimistically, you might see some kind of "choosing a chief by consensus" type of situations that you see in some small hunter-gatherer societies (being careful with the word "savage", which is... usually misleading). It'll depend a lot on the size of the crew.
There is a lot of precedent for this. Even on Earth, in 2026, international maritime law states that there is no such thing as a vessel with "democracy" and that a captain always has supreme command. Ships, airplanes, etc are all in a category that operate as strict autocracies.
Sure.
How long's the longest voyage these days?
Mutinies aren't so common nowadays, but they were when ocean voyages were measured in months and years.