Is it permissible for a sister who is not in need to work in a mixed-gender hospital, wearing a white lab coat that reaches mid-thigh with trousers underneath, and wearing the niqab? And is a brother driving his sister to this mixed-gender workplace considered dayyathah (cuckoldry), or is it choosing the lesser of two evils?

A woman is not permitted to work in places where there is unlawful mixing [of genders], nor is she permitted to wear trousers in front of non-mahram (definition) men. The niqab that shows only the eyes is permissible, but the one that shows what is below and above the eyes is forbidden.

It is not permissible for a man to drive his sister to a job where there is unlawful mixing, because it is an aid to sin. However, if not driving her would lead to her being alone with a driver, then he should drive her in that case, committing the lesser of two evils. His action in this instance would not be considered disinterest [in the family's honor] as long as he disapproves and advises her.