But for the expectation to exist that particularly the guy do it for the girl, this idea comes from outside cultures (b’cholkoseihem).
I find it so curious that whenever a guy wants to find a “legitimate” excuse to not act like a gentleman, he refers to the “bechukoseihem lo seleichu” card. The fact that Non-Jews do or do not do a particular mannerly thing for a woman, in no way makes that mannerly thing THEIR thing. Good manners are , or should be, universal. If men are to be considered as “stronger” than women, or their protectors, then it makes total sense that they should likewise treat them deferentially with good manners. Not to be outdone by the men, good manners are incumbent upon us ladies as well. If I see someone is behind me as I go through a door, I hold it open for them. I saw a lady on the avenue the other day struggling with a door and a baby stroller, and I ran up to her and grabbed the door to help her. That is just plain doing the right thing.
If you are going to find halachic reasons to justify one’s acting in a boorish manner, I imagine Hashem is NOT pleased.