Is it forbidden for a wife to remain in a mixed-gender workplace as long as she does not neglect her household duties, and is she obligated to inform her husband or seek his approval before disposing of her private property?

A woman is permitted to go out to work, provided she adheres to Islamic guidelines. However, she is not allowed to go out to work without her husband's permission, except in cases of necessity or need. She must obey her husband and quit her job if it involves unlawful mixing (of genders). The reasons you mentioned, such as stimulating the mind or alleviating loneliness due to the husband's absence, do not justify working without his permission. If you specialize in electronic work, you can look for work in this field from home. A woman has her independent financial control, and she can dispose of her money as she wishes in permissible ways. She is not religiously obligated to seek her husband's permission, though seeking it is good.