Is the money my friend earns from her work at two companies, with official appointments—one where she works one day a week and the other for the remaining days, with the knowledge and support of her managers—lawful or unlawful?

There is no religious impediment for a Muslim to work two jobs or with two entities, as long as they fulfill their work completely according to the agreed-upon contract terms. The government is not allowed to prohibit this except for a legitimate, recognized interest. The worker is not permitted to be absent from either of their jobs except with the permission of the employers, nor to collude in absence without the authorization of those empowered to grant it, as this constitutes betrayal and deceit, and a violation of Allah Almighty's saying: "O you who have believed, fulfill [all] contracts," and the Prophet, peace be upon him, said: "Muslims are bound by their conditions," and "He who deceives is not of us." If your friend's absence was without permission, or if the prohibition was for a legitimate interest, then it is not permissible for him to do so.