Is a man considered a Dayyuth if his wife does not wear the Niqab? And is he considered a Dayyuth if his wife works with him in an "open space" administration, wearing the legal Hijab, modestly, and not mixing with men except out of work necessity?

A Dayyūth is a man who has no ghīrah (protective jealousy) over his family and womenfolk. A man who requires his wife to wear the Shar'i hijab, but without covering her face, is not a Dayyūth, because the opinion permitting the uncovering of the face and hands is a valid one. As for a woman's work, there is no harm in it as long as she maintains the Shar'i hijab and adheres to Islamic manners, such as lowering her gaze and not being alone with men. She should also have a separate workspace where she does not mix with men unless it is necessary for work, especially if her husband is with her.