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The Royal Navy never was -quite- as aristocratic as the army, in which one might simply buy a commission if one had money. Navy officers needed specific training for which a specific inclination and talent was particularly prized, no matter how blue your blood was, tho it might incur you preferential treatment, not guarantee you a command. You did need to pass the examination for lieutenant before a board of captains to become an officer, and that could not be bought. Captains also had the freedom to take on whomever they pleased on as midshipmen. There was a clear incentive to taking on those who had letters (incurring favor with those up the chain of command to make Post, for example), but a less fortunate Captain, say one that might have a few debts might take one as a personal favor, or might take his own child, or the child of a career seaman as form of recognition and thanks for his service.
Oh! Midshipmen actually are taking lessons in navigation and mathematics from the sailing master and bookkeeper/purser. Sometimes the Captain himself oversees lessons to pass time on long voyages. The point of these lessons is to prepare the midshipmen for their examination for Lieutenant when they reach 18 years and 6 years at sea, guaranteeing a steady supply of competent officers. So as a midshipman you'd not only be allowed mistakes, you're not even expected to have experience. If you want a good example of a midshipman's life, look for a Mini Series called "Hornblower". Very good presentation of sailors and officers' life shipboard and a great source for learning the jargon. There is also a very young midshipman in the movie "Master and Commander". Both fantastic :)
Oh she's totally a go, not a moment's hesitation.
If it helps, in the Royal Navy there are two or three positions a 15-16 year old might hold. Indeed the base seaman is one of them, but there also the powder monkeys (generally even younger), the steward/cabin boy, and there are midshipmen, which are a little more interesting if you'd care to do a little forgery with that "letter of service", but would actually get you trained in the finer arts of sailing a ship. Whatever you choose, I'm sure you'll do great!
If she's thinking of singing up with the navy (the ship is Navy until the mutiny), her charismatic personality probably won't be the saving grace she needs. She'd need to pass-off as a man, at least until the officers are dealt with. As a passenger however, she could have indeed run off to make a life for herself in the colonies. After that point, I doubt the pirates would be able to refuse her anything. Like... ever. ;)