Why is the mixing of men and women permissible in Hajj and Umrah, while it is considered forbidden in other contexts? And did Lady Khadijah mix with men due to the nature of her commercial work?

The mixing of men and women is not absolutely forbidden. Rather, what is forbidden is that which leads to fitnah (temptation/strife) or lacks proper Islamic controls, such as khalwah (private seclusion), speaking in a soft and alluring manner, and displaying adornment. This forbidden type is never permissible. As for the work of Lady Khadijah (may Allah be pleased with her) in employing men in her trade, this does not indicate forbidden mixing. Rather, she would hire them to conduct trade with her money, and this does not violate Islamic law.