Is it permissible for a Muslim girl who works as a nanny in a house that has a dog to pray, knowing that she will sleep in this house and has no other job opportunity?

It is not permissible for a Muslim to hire himself out to a Dhimmī (a non-Muslim living under Muslim rule) for service, because it entails the subjugation, humiliation, and exploitation of a Muslim. The prohibition is even stronger if it leads to non-compliance with Islamic regulations, such as hijab, avoiding khalwa (private seclusion with a non-mahram), and handshaking. As for praying in a place where there is a dog, the prayer is valid if the place is not touched by any impurity from the dog. If it is touched by impurity from the dog, the Hanafis and a narration from Ahmad hold that it becomes pure by sun, wind, or dryness. In contrast, the Shafi'is and Hanbalis hold that it does not become pure except by pouring water over it, and extensive washing is sufficient.