I am a woman, I work in an executive position for an international company. I always smile and say thank you when someone holds the door for me.
I have had great success by implementing “if you want to be treated equally you have to act equal.” I was fortunate early in my career to have mentors give me advice on how to successfully navigate being a woman in a male dominated business.
And for the pay gap, I believe part of it at least at my level, is that women don’t negotiate their compensation packages nearly as often as men do. One of my male mentors told me to never accept the first offer even if I’m happy with it. Negotiating for pay makes me uncomfortable, but I have never had a company reject my counter offer to their salary/benefits-package.