Will the husband be sinful if his wife works in a mixed-gender environment without his consent and without her needing the money? And will he be sinful if he divorces her for that reason, given that they have a child together?

A wife is not permitted to leave the house to work without her husband's permission, especially if he has stipulated that she not work. If she does so, she is sinful. Divorce is not immediately recommended; rather, the matter can be addressed by speaking with her family or by following the methods for addressing nushuz (recalcitrance), which include admonition, abandonment in bed, and light, non-bruising striking. If she does not respond, then divorcing her becomes permissible. As for her working in a mixed environment with impermissible intermingling, her sin is magnified, and the husband must prevent her from doing so because he is responsible for her. Attention must also be paid to the specifications and conditions of the شرعي (Shar'i) hijab (definition).