I am not certain why "o chapeu" translates to a possessive "his hat". Anyone know?
Maybe because the sentence makes it almost 100% sure that the hat belongs to the man, so in English his makes more sense. Anyway, using the should also be accepted!
Is that pretty common in Portuguese to use "the" instead of the possessive pronoun when the sentence makes it clear whose hat/coat/shoe/etc it is?
Yes, pretty common! For example: o homem estava com A mão no (em+o) bolso = the man was with his hand in his pocket ;)