I try and do things for my friends and family. I never go out of my way to help women more so than men, but I do tend to do things for people who smaller and/or older. I'm also not straight, so I the things my partner finds gentlemanly in our relationship is different. I do a lot of basic chores and things like that but only because I usually have more time and am home earlier. I tend to let the husband go first in doors or trains, i usually serve him first with food and drinks. He does a lot of the heavy lifting, but that's because he enjoys manual work and has more upper body strength.
If I hold the door for someone, I typically don't even notice if they are male or female. Being from Alaska, the ideas of being a gentleman is very different. I can't even imagine trying to cater to women up there in a way that most people consider "gentlemanly"; Alaskans are pretty damn independent, and that goes doubly for the gals.
In Japan, people tend to think I'm a perfect gentleman because I hold doors or let them go first. People think that folks here are super polite but their idea of polite is much different. I held the door for a girl and her boyfriend once and she told her boyfriend its because I was flirting.
I just treat people politely and the way I would want to be treated. Doesn't matter if they are male or female. Being a decent human was hammered into my head when I was a kid.