Your friend doesn't seem to understand what it's like being a woman walking alone at night. The thought was nice but he failed to realize that what he was gonna do was going to make her feel more uncomfortable than just being out in the rain without an umbrella.
As far as treating women extra nice, I do it sometimes, not sure if it's sexist, but I'm still being nice to people in general - hold the door, hold the elevator, etc. But if we're two people in an elevator both getting out at the same floor, if I got in first, I exit the elevator first, unless it's a woman, in which case I let her go first regardless of who entered the elevator first. Might be sexist, but, at the end of the day, it's not like I'm being an asshole to men and a nice guy to women, so I don't feel bad about it.
It's funny you say that because I prefer to leave the elevator first especially if a woman is there with me. The reason is because one time I did the polite thing and let the woman go first. She turned in the same direction as my apartment so I was following behind her. She kept looking over her shoulder and then stopped and said "why are you following me". I told her I wasn't following her and my apartment was right here. I put in my key and went in. So yea being a big Hispanic guy means I don't let women go first.