Is the prayer of a Muslim woman who lives with her sister, who owns a dog that roams all over the house and whose saliva gets everywhere, accepted, given that she cannot remove the dog or designate a special place for herself? And what should she do in this situation to ensure the purity of the place?

We believe it is better not to live in a house with a dog, because angels do not enter a house that contains a dog, and to avoid doubt and whispers concerning purity and prayer. However, if living there is unavoidable, and you pray in a place whose purity you do not know, the default is that the place is pure, and its impurity should not be ruled except with certainty, in accordance with the rule: "Certainty is not removed by doubt." The Companions used to pray in the Prophet's Mosque, and dogs would enter and exit it, yet they did not sprinkle anything. This is supported by the hadith of Ibn Umar in the Sahihayn, and they used to pray based on the default, which is that the place is pure.