Is it permissible to ask about the details of a previous engagement and a girl's life before marriage, and does compromising on the required level of religious commitment for the sake of a calm personality and good manners contradict the hadith: "So, win the one who is religious"?

There is no need to ask the girl about the details of her past life or her previous engagement, for you have no right to that. Rather, you should conceal and overlook. What is apparent from her goodness and righteousness is sufficient for you, and if she is so, then rely upon Allah and proceed with her proposal. You can seek the help of her brother to compensate for any shortcomings in her knowledge or work during the engagement period.