Of course it is sexism.
In the first half of the 20th Century, women expected men to hold doors open for them, give up chairs for them, pay for everything, etc., and women were expected to stay at home and keep up the house.
In the 21st Century those expectations are out of date and should be forgotten.
I would NEVER give up my chair for a healthy woman.
I have never stood when a woman entered a room, because to me, everyone should be treated the same, regardless of sex.
(Actually, I don't think I've ever seen that situation at all.)
I respect men and women, and if I would not give up my chair for a man, why should I give it up for a woman?
The only time I would give my chair up at all is in the case of a feeble, old, ill, or pregnant person, or if I'm on my way out of the room.
I also only open doors for people who are behind me, and most of the time, they are so close that they have to grab it themselves to walk through it. It's not because I want to be a 'gentleman', it's only because I don't want the door to hit them in the face. If they're in front of me, I'm not going to rush in front of them and open the door for them, that's just foolish and unnecessary. Now if it's an old, feeble, ill, or pregnant person, yeah, I'll open the door for them, or if someone's got their hands full, but if they're perfectly capable of opening their own door, nope, get it yourself.