Is a wife considered disobedient (nashiz) if she mingles with and treats men in her work after her husband approved her travel based on her assurance that she would not intermingle, and is the husband sinful if he divorces her because of that?

If the wife’s work involves unlawful mixing between men and women, then it is not permissible for her to continue in it, because it is a cause for fitna (strife/temptation) and corruption. She must leave the work by her husband’s command. If she does not, then she is disobedient (nashiz). If she has lied about the nature of the work, then both she and her sister are sinful, and they must repent. The expulsion of the wife’s sister’s husband should not occur merely because of your insistence that your wife leave her work.