It's not that simple. There is a script. If I don't go when I'm meant to, then it gets really awkward. If I break from the script, no one knows what to do. They awkwardly wait for me to finally move, and if I don't, they eventually get clearly frustrated as they step in ahead of me.

Try saying "I'm not going in" or "I'm not going in right now". Idk, maybe I'm interpreting the situation incorrectly.

Not in the same way. A woman might hold the door for a group of people or if if I'm running towards it and she's already there. But if it's just her and I standing at the lift for example, she won't awkwardly wait for me to get on first, but a guy will.

I can't help, but wonder if this is a regional thing. I live in the southern U.S., and everyone holds the door open for everyone. I even see men holding doors for men.