Is Chivalry sexist? Yeah it is. Why because the mentality that lead to specified gender roles that have come with Chivalry contributed to a lot of the gender issues we are currently dealing with. Women are to be coveted and treated like objects for men to take and men are to be the heads protectors, essentially guard dogs, to women. It's a system that has lead to a lot of stress on both genders. More so for women I will say.
If you're going to do something nice for someone like hold the door you should do it for everyone. If you would offer your seat to a woman you should do it for everyone.
Now that being said pretending that men and women are the exact same isn't helpful either. Now I'm not talking in a sense women are inferior to men as a gender. I'm saying biological things like a woman is more likely to be weaker than a man on average. Flip side women are more often more flexible. Women deal with biological issues that I as a man will never have to and vice versa. As well as cultural things, women are perceived to be easier targets and therefore are often attacked more than a man.
So I might be more inclined to ask a female friend if they want walked to their car late at night than I am a male friend. Now I would still base this on the individual person but I still understand that there is a difference between a man and woman in this situation most of the time.
What I'm saying is that we are all people. We should all treat each other as people first but understand we are different and be okay with that. I do not feel Chivalry as it has become does that. It puts our genders first and sometimes as the most important thing while seeing us as people second.