Halacha is precisely the opposite. Halacha is men first. Halacha specifically states as such even when saving lives.
When a girl says “ladies first” to obtain priority for herself, she is a victim and perpetrator of the American and Western self-entitlement culture who subscribes to non-Jewish ideas and culture.
Why should a guy be any more inclined to open the car door for his date than he would open the car door for his younger or older sister? She, at least as much, ought to open the door for him.