I believe whomever asks a person out should be prepared to pay for the date. I can't imagine asking a woman out expecting her to pay.
Having said that if I received a call inviting me out to lunch or dinner I would be offended if the woman (expected) me to pay. I'm not a human ATM machine. Once again whomever extends the invitation should be prepared to pay (in my opinion).
I've read some women prefer to pay their own way to remove any "ideas" a man has that causes him to think the woman is somehow "obligated" to go beyond where she wants to at the end of the date.
We as a society have always equated dating/money with sex and how much one cares. The phrase: "I'm not a cheap date!" implies don't take someone to McDonalds for a "Happy Meal" and expect to have sex afterwards.
On the other hand some women would have a difficult time accepting a date to a 5 star romantic restaurant and a play with a guy they're not attracted to while other women would jump at the chance with no intention of ever going out with the guy again.
Men tend to only ask women out whom they (do) find attractive.
As for opening doors I do that regardless of gender for the most part especially if I'm walking into a building and I'm aware there is someone right behind. I'm fully aware some women would rather not have a man standing there holding a door open for them when they are still (several feet away) therein forcing them to almost run.
Nevertheless it always seems odd to me when people either refuse to say "Thank you" or acknowledge kindness of any kind.
I guess some women don't want to feel "compelled to speak".
They'd rather open their own doors, lift their own bags, pull out their chairs. Essentially some people see chivalry as something born out of sexism. Women were viewed as the "weaker sex" who required protection from men and needed to be "looked after" much like children. There's a battle between feeling "special" or "equal".
Ultimately I believe it's better not to try and guess which type of woman you're dealing with but instead just be true to yourself.
In the end everyone wants to be loved and appreciated for who (they) are.