I've thought of myself as a feminist ever since I learned the word at about twelve years old. By that, I mean only that I think men and women are equals, are equally capable of almost everything, and should have equal rights under the law. It doesn't mean being mannish, and it doesn't mean not doing or enjoying things that are traditionally feminine. It does mean that I can wear jeans or dresses, spend time quilting or woodworking.
When two people are going through a door at the same time, whoever is in the most convenient position to hold it open should do so for the other person, regardless of gender, IMO. There are exceptions, such as disability or when one of the people is laden down with packages. It's very annoying when a guy pushes his way in front of me just so he can hold the door for me. Or standing in the doorway so that I have to squeeze past him to get through. I've been yelled at and called names when I've gone through an adjacent door instead or asked the guy to go through instead of squeezing by him. Where are the manners in that???
OTOH, when a man knows how to do it right and does it with that certain something that lets me know he's doing it as a gentleman would for a lady, I find that delightful. (Then again, I don't have the patience to wait for someone to walk around and open a car door for me. I'd rather just get out myself.)
The whole idea of treating a woman like a lady is kind of formal. Overall I'm a pretty casual person and wouldn't require that in any relationship. But I think that's a matter of individual preference. Treating me with respect and decency is a whole other thing, though - that is required.
And it's reciprocal. Women can't demand or expect to be treated like a lady or with respect without doing the same to the man (uh, treating him like a gentleman).
As far as what happened with your daughter, the guy was wrong. If the cousins wanted a night out, they should have arranged it beforehand. Since he had other plans, he should have kept them. I doubt this was a case of wanting to spend time with cousins not seen often, since he could see them the next day. Or arranged to meet up with them later on, or with your daughter, etc. This sounds like guys' night out to me.
It's up to your daughter to decide whether she wants to be treated like a lady or less formally, but she does need to decide what the limits are and let him know.