I don't think I would ever hit a girl. I would definitely defend myself from one, and it includes hitting, if they assaulted me first, but I would never just outright hit a girl/woman. I have started many fights with guys, throwing first punches and all that, but I won't do it with a female because I'm not comfortable with the idea of it, no matter how sexist, or old fashioned it seems. I will say I am not sexist, I do not believe women are inferior to me, I know a great deal could kick my scrawny butt.
I'm a guy, and I'll fight with a guy because I know how they pretty much work, but girls are unfamiliar territory to me, I don't feel natural even thinking about hitting one. It's not that someone put the idea that hitting girls is wrong in my head, it's simply that I'm not comfortable with the unfamiliarity of it. Unless they are attacking me, I won't hit them. If they get in my face, I'll walk away. I might not do the same for a guy, and it doesn't sound fair and equal but I just don't care.
And yeah, I do hold doors open for women, men too, but I usually find myself doing it more for women. I don't go out of my way to do it, but if I reach the door first and there are people behind me, I will hold it open and let them go first. To me, it's just polite, because I'm certainly not interested in seducing a girl
Besides, if something horrible was on the other side of the door, say the edge of a cliff, letting them go first would have warned me. It's logical thinking.