ujm-I assume a bochur would consider it quite odd for the girl on the date to go over to driver’s side door and open the car door for him. And it would surely throw him off if she would say “could I please have the keys to the car so I can open the door for you”……
Women are quite capable of opening the car door, and I would agree that it is probably stemming from chukas ha’goyim, but I don’t think the bochur is b’sheeta opening the car door because he wants to be like his neighbor the gentile. He’s just trying to be a mentch–and yes, no need for him to watch her get in etc…..
I had no problem in my dating years with the bochur not opening the car door for me. It was just frustrating when he would forget to unlock the door and I’d have to knock on the window. (nope-didn’t marry that one, but for other glaring reasons)(I guess part of this post can go in another thread too.)