Yes, a common age for young officers to enter the Navy was about 12 or 13 as "volunteers" or "captain's servants" followed by promotion to midshipman by 13 or 14, see Nelson for example. Some successful officers joined when older but that meant 17 or so, see Cochrane.

In Patrick O'Brian's books there were several instances of young midshipmen getting wounded and having amputations. It was not uncommon for a severe limb wound to be amputated especially if infection set in.