Is it permissible for a nurse to travel by plane with patients—women and children—without a mahram, especially if there is additional pay for it, and would this money be considered unlawful if traveling without a mahram is unlawful?

It is not permissible for a woman to travel without a mahram, due to the saying of the Prophet (peace be upon him): "It is not lawful for a woman who believes in Allah and the Last Day to travel the distance of a day and a night except with a mahram." The fare taken in exchange for traveling without a mahram is unlawful (haram).