All the Gentleman stuff has function behind it. It streamlines things between men and women especially.
If it's understood that the man will be a gentleman, the women can fit seamlessly into the flow of things. Man opens door, woman sails through. Man draws back her chair, she seats herself first and with smooth efficiency. The woman gets the seat that faces into the restaurant. The man always walks on the outside of the sidewalk, closer to the traffic.
And a hundred other things, which I hope you will add to in the comments. It is amazing how smooth things get when everyone knows what's supposed to happen; and I will always believe almost all women love to be protected, cherished and appreciated in this way.
Old ladies (your mom and her friends) are often astonished by and grateful for this kind of treatment, and it's easy plus points for you.
Some other aspects: when a friend shows up beside your table at the restaurant, stand up and shake hands. Always introduce him to whoever you're with. You don't have to invite him to sit with you, but it shows respect and doesn't leave out the third person at the table. When you greet a friend, use his name. I don't always shake hands, but more often than other men I know. I like it. It's physical evidence that you have respect; and gestures of respect are never a bad idea.
In modern times, being a gentleman is about smooth navigation and affirming respect.
I'd be glad if the men who've come here will chime in with what's important for them. What I've said is just the tip of the iceberg. I believe it to be a very worthwhile study and that it builds a strong society.