Eraser_, this is hardly the "latest craze" -- feminists who were around in the early 1970s have written that it was a big controversy then as to whether accepting the politeness implied that you were too weak to open doors for yourself or not. Me, I have no problem with anyone opening the door for me if they get there first, but I'll open it for them if I do. (I once had a boyfriend who would do all sort of running and dodging around me to get there first, which seemed like overkill.)

My rule is that it's sexism if you would treat all members of one sex a certain way, but never any of the other sex; it's politeness (or rudeness if it's a negative treatment) if you would treat people of all sexes that way; it's flirtation if you would only treat attractive/interesting members of the appropriate sex that way.