Why is it the man who is supposed to hold doors for a woman and to let them go first?

Laws don't always determine why we do things. I don't think men need to let women go through first. I always hold the door as I go through and let people grab it. I don't want it slamming in anyones face. This is all I expect out of other people. I don't need men to hold it for me and wait for me to go through first. I appreciate the kind gesture but its not necessary. I don't need special treatment. The only times people need to hold it open and let a person go through first are for old people who could use the help, someone with a stroller, someone with full arms etc. 

Actually, men are supposed to enter the room or building first, (as they can check if it might be "dangerous")
and they are supposed to hold the door from the inside.
And why hold the door? Because women always had these massive dresses, they couldn't have opened the door themselves.

But the main thing is what most people get wrong, is that men should go in first so the "the bottle hits them" not the lady.

Yes, there is. It says, "Age before beauty". I must admit, however, that when someone opened a door and waved me through, saying that, I replied, "Pearls before swine", and walked through.

That is the olden days when women were weak and men had to do everything for them. It also meant a woman had to wait for a man to open the door. Feminists didn't like that. Women are quite capable of opening doors. So nowadays women don't appreciate chivalrous men. They will often be angry too. So don't do that anymore unless the person is old and weak or carrying something.

Custom is not law, this is a strange word to the past few generations, but you can look it up on your own so you won't think the alien shape shifting lizard people are behind it. The word is Manners.