WELCOME ABOARD!
Real life on a ship:
All seamen are petty officers in training. They may be knot-tyers and they may be radiomen or machinists, but they are learning a specific skill. Once the reach the highest (Seaman 1st Class for the US and Matrosenobergefreiter for the Kriegsmarine) the become a junior petty officer and are considered 'rated', officially a member of that job description.
In real life it is possible for a sailor to go from sailor to petty officer, but in real life the captain doesn't have the power to promote him. He can recommend him for promotion, but it is done through official channels. And there is a reasonable chance that he would be reassigned at that point.
A Chief Petty Officer is the highest an enlisted man can go. There are 'higher' ratings, at least in the US (Senior Chief, Master Chief), but if someone is considered qualified for promotion to actual officer rank he has to go back to Officer Candidate School (or the German equivalent) and undergo training, so he would disappear from your crew and be replaced, not promoted on your boat.
Yes, there are battlefield promotions (one of my favorites is in
The Longest Day, when Robert Mitchum promotes Jeffrey Hunter from sergeant to lieutenant and sends him of to die just minutes later), but they are usually considered 'brevet' promotions, which means they haven't been officially confirmed. 'General' George Custer was officially still a colonel when he was killed.
But that's the army and that's a battlefield. It's not nearly so common in the navy, where rank means less than proficiency.