If you're a gentleman, you are polite to both genders: you merely reserve certain types of polite behavior for the opposite sex. There's no reason to claim it's because women are considered inferior; it could be considered along the lines of competitive marketing in the "seller's market" of men if one had to, though. For me, treating people equally doesn't demand pretending that both genders are the same and treating them both rudely.
Call me old-fashioned, but I believe that you offer to take a friend's coat at the door and offer a lady your coat if she's cold: there are biological reasons for her to feel colder than you anyway. It's the same courtesy you extend to breaking up physical fights between male friends. In both of these cases there's a sort of stereotype to initial state expectation, but a true gentleman is also going to do what he can to break up a physical fight between women (shut up, you) and offer his coat to the guy stupid enough to wear a sleeveless shirt.
You hold the door open for everybody, regardless. It's just what you do.
