What is the ruling on working in mixed-gender hospitals when no alternative is available, given the physician's necessity to interact with unveiled women and the difficulty of lowering the gaze? Is working there considered a continuous sin, or only upon looking? Does this affect the salary?

If mixing (between genders) in a hospital is forbidden, then it is not permissible to work there except out of necessity, such as the need for earning a livelihood and the unavailability of other work. Likewise, it is not permissible for you to perform surgeries on women unless it is a necessity, and if another woman can perform it, then it is not permissible for you to do so. Sin is negated only within the bounds of necessity. You incur sin by working in a forbidden mixed environment without necessity, or by letting your gaze wander. Working in a mixed environment does not affect the lawfulness of the earnings, as long as the work itself is Islamically permissible.