At what age does a boy in Westeros legally become a man?
16
And how does age effect the enforceability of oaths? Specifically, if Jon Snow was 14-15 years at the time he swore his oaths to the Night Watch can he avoid them as he was, maybe, a minor?
No loopholes. Once you say the words, you're in.
That being said, the Watch would not give the oath to a boy that was seriously minor, like a 12 year old.