Is it permissible to hold a wedding ceremony in a mixed hall (men and women) with quiet music, especially if that is a condition for the marriage to take place, and would the son be sinful for agreeing to that?

What the bride's family requests regarding the presence of music and the intermingling of men and women is impermissible. As for music, it is forbidden (haram). As for intermingling and displaying oneself (tabarruj) before people, it is among the grave sins (kaba'ir) due to the presence of qawadah (pimping/cuckoldry), tabarruj, unlawful gazing, and the arousal of desires. Therefore, your son must not obey them in this act of disobedience, even if it leads to the breaking of the engagement. He must establish his home upon the fear of Allah and His good pleasure.