Pondering the Preposterous...
But I don't see anything wrong with that particular courtesy. As long as the man isn't grinning smugly as he holds the door, or the woman isn't feigning helplessness as she waits for the man to perform his "duty." I had a History professor who on the first day of class made it very clear that she would not tolerate men in her class who displayed such "chauvinistic" behavior.
I'm sure she saw plenty of doors slammed in her face that semester.
So here's one gender role, though more of an expectation, that I find utterly stupid. And that's for a man to stand up while peeing. Really...if you stop and think about this you're likely to laugh at how obscene and lame it is to whiz over a toilet bowl and hope your fluid hits the target. So, come on, if there's a seat, use it. Pity the poor individual who has to clean up after a man who can't aim.
There are exceptions, however--two that immediately come to mind. If it's a public restroom, by all means, stand tall and do your best William Tell. In that respect we men are very fortunate. And, of course, if you're out in the wild it's perfectly acceptable to pee in the vertical position. Especially if snow is involved; I would never turn my back on Mother Nature's very own etch-a-sketch.
But if you're in your own home, sit down and enjoy it--you'll save in the long run on toilet cleanser. And don't complain about the seat being cold--you wuss! Just think about the poor schmuck who has to wipe away your golden dribble.