When I am walking with somebody, male or female, and I happen to be a step or two ahead, I open the door and hold it for them. Sometimes I go in first and hold the door behind me, but sometimes I stand aside and let them go in first. I don't know what determines which I do -- it might be the gender of the person, but sometimes it's just random.
When I happen to be a step or two behind, I'll generally pause a bit and let them open the door. It just makes sense.
When we are walking shoulder to shoulder, there is always a bit of awkwardness. Who opens the door for whom? The person closest to the handle? The person on the right, since most doors open right? The male, if there are two genders?
When there are two doors, I'll always open the door in front of me at the same time the person I am with is opening the door in front of them. To me it makes sense. Two doors, two people, and we can keep walking in step if we each use our own door. This doesn't seem to be common behavior. I've noticed the majority of people like to use one door, half-lining up, each walking through and then holding it behind them for the next person. While I think the polite "working together" aspect of that is cool, it still seems a bit inefficient.
Me, I'll walk past the group and head for the other door. (It's always so annoying when it is locked. For some reason that is sometimes done. It's a door people, and it was born to be opened and closed. If you're going to let us use one, let us use both.)
However my other-door-using approach leads to some awkwardness too. Like when a guy pauses to hold the door open for me, and I'm already heading through the other door. I do feel bad when that happens, and I generally apologize. I'm not trying to be rude or overly feminist or anything; it just frequently doesn't occur to me. Like I said: two doors, two people. So long as the two people are equally able (healthy), it just seems most efficient to use both.
I think this bothers men, and I can think of at least one ex-bf who has expressed this. But not to me, he joked about it to other people with me around. Nobody that I remember has asked me to let them open the door, but I probably would let a bf do that if he asked. At least I would try harder to notice such occurrences. I wonder if it was really an issue, or just a joke?
I've started paying attention and trying to sense when it bothers men -- intentionally slowing my pace so that he will be in front and can smoothly hold the door. It still seems ridiculous to stop and wait in front of the door for him to open it, but if I can anticipate the situation and slow down ahead of time, it feels more natural. If it makes somebody happy, I'm willing to do that.
I remember one ex who really wanted to open the car door for me. Now that had never even occurred to me! Unless I have a whole pile of stuff on my lap, or am quite sick/injured, I just open the door when we arrive and get out. However apparently he had always tried to do it, and I'd just obliviously gotten out before he could get there.
So we arrived one day at a restaurant on a date, and he was determined to open that door. So we parked and he hastily walked around the car. Unfortunately I saw his speed and thought it meant we were in a hurry. So I hurried too and leaped out of the car, ready to rush off to wherever we were going. When he told me what he'd been trying to do, we both had a good laugh at the situation.
The fact is that I really don't mind a man holding a door for me. It's gentlemanly and traditional, and I certainly don't get offended. I'm just so used to opening my own doors and getting through them in the most efficient way possible, that I often don't think about it when it's happening.