Sinbad was already being pushed away from Proteus before Marina came along.He sees her on a boat and falls head over heals to the point that he leaves everything and everyone he's ever known because he realizes that she's not meant for him? Just a bit too convenient. It would make more sense that his relationship with Proteus has been strained to the point of snapping—after all, Proteus is a prince and he is not—and Marina's arrival was just the straw that broke the camel's back. She epitomizes all that he's been realizing about class and station, as well as the distance between himself and Proteus, and so, he leaves.
Kale's actually the one running things.He must have pretty much raised Sinbad after the boy ran away from the palace, and while Sinbad is fearless, a good fighter and captain, Kale has the muscle and fear-factor that must have been necessary to command a ship of pirates. He's just too nice to let Sinbad know who's actually in charge.
Marina's famous, and the crew already knew about her before the events of the movie.They let her on the ship, no questions asked, and are quite nice to her throughout the movie. Sure, she has jewels, but it would make more sense for her to be renowned as an ambassador of some sorts, perhaps a policy-maker who fights for the rights of the poor. They know her face and name due to the good changes that she's implemented, and hence are much nicer to her than they would be otherwise. MacGuffin, it's quite clear the book does something—witness the enormous aura/field it constantly projects when opened. While it's entirely possible it creates some sort of aura of prosperity, the name implies more than this. So to use Dungeons & Dragons terminology, it projects the same sort of pacifying field that an Apostle of Peace can create, only writ large: those who enter it are affected as by Calm Emotions, having their negative emotions (but especially rage and violence) suppressed. This would explain why, extended disarmament and friendship with Proteus nothwithstanding, a known thief who had just tried to steal the Book was allowed into the celebration at Syracuse—because Dymas and the rest of the Council of the Twelve Cities knew he couldn't steal it while within its influence. (Note that on the ship, although he claimed he was going to steal it, we don't know if he could have gone through with it because Cetus intervened. And it is likely that taking violent action in order to defend yourself or others, as Sinbad, Proteus, and their crews do, is an allowable exception to the field.) This would also explain why immediately after Eris steals it, the Council goes right into imprison/execute mode, and why she would want it stolen since it'd be very hard to accomplish anything chaotic within its influence. As to why she could steal it, because it is only a minor artifact (or else was made by a deity of a lower divine rank than she is) and thus as a goddess she is unaffected by it—to a point, since it's also possible the cloud, earthquake, and cracks were a result of her own chaotic aura being able to reassert itself once the Book was closed as much as by the simple absence of the Book's aura.
Sinbad (and by extension, Marina as well) is an ancestor of Tony StarkHe's exactly like Tony, down to the snark, the It's All About Me tendencies, the hidden heart of gold, his worries about being a Jerk With A Heart Of Jerk, being very much an Unlikely Hero and falling for a woman who quite simply will not put up with his bullshit. On top of that, he's got the same looks, the same fondness for the colour red, even the same facial hair!