My "women first" policy generally only applies to female friends I feel "protective" of, or a significant other.

I will hold the door, out of common courtesy, but not allow them to pass in front of me, just hold my arm out behind me as I pass until they grab the door. I do this for all people in general.

I don't treat women with more or less respect than I treat men, unless said woman is my significant other or just a close friend.

However in relationships or in dating I'm generally a "you go first" or "hold the door/open the car door" etc type of guy.