Is it permissible to keep a dog for guarding inside the apartment of a woman who lives alone? And do angels refrain from entering the home due to the presence of the dog? And is the prayer affected by that? And does the ruling differ if a room is designated that the dog does not enter?

It is not permissible to keep dogs except for guarding (a hunting dog, livestock, crops, or a home), and that is out of necessity. If the need is pressing and there is no alternative, then it is permissible to keep one. It is preferable to allocate a place for it in the house. In this case, it does not prevent the entry of angels; rather, a dog that is permitted does not prevent the entry of angels.