I remember when I was living in a student household, having the toilet seat debate. My argument was that four guys out-numbered two women, so the next person was more likely to need the toilet seat up, which meant it was basically more practical from an efficiency point of view to leave it the way it was when you were finished. Since then I've just got old and decided it just looks better down, so wherever I am I put it down out of habit. 
As for giving up my seat, I'm gender neutral. I guess statistically speaking women are more likely to be either pregnant, struggling with children or less strong for carrying heavy bags, etc. but I wouldn't give up a seat for them just because they're women.