I didnt read a single reply. Apologies if I am repeating anything.
The gemara states that a man should not walk behind a woman. I dont remember if this was a middas chassidus, a halacha or just an eitza tova. Whatever it is, there is certainly legitimacy to not walking behind a woman. When I was dating (20 years ago), I was advised to hold open the door for the girl I was with, but, to wait 3-4 seconds before entering so that I wouldnt walk in directly behind her (and the possibility also existed that once inside she would go a bit to the side and wait until I walked in too, making it so that I was not in back of her – in fact, several girls I dated walked through the door, directly to the side of the entrance so that when I walked in, they were not in front of me at all – they were well aware of this gemara too).