Is it permissible for a woman to work as a doctor in a mixed-gender hospital, given the nature of the job which requires dealing with male patients or colleagues, while adhering to Sharia guidelines such as wearing the hijab and avoiding makeup and perfumes, and trying as much as possible to limit her interactions to women?

The fundamental principle is that it is not permissible for a woman to work in a place where she mixes unlawfully with men, except out of necessity or severe need. Similarly, it is not permissible for her to treat men except out of necessity or severe need. If a woman feels secure from temptation by working in hospitals where unlawful mixing is prevalent, and adheres to the Shariah guidelines as much as possible, then there is no blame upon her, Allah willing, due to Allah Almighty's saying: "So fear Allah as much as you are able."