Must one avoid the cat during prayer if it has been afflicted with impurity and the impurity itself has vanished but has not been washed away? And is it true that the default ruling is purity regarding shed cat hair, which may be from an impure or pure source?

If the impurity that affected the cat is removed with pure water, or with one of the pure liquids according to one opinion, then it becomes pure, and whatever it touches or whatever its hair falls upon becomes pure. However, if the impurity is removed merely by dryness or sun, the cat does not become pure by that. The impurity is not transferred merely by the contact of the cat's impure hair with the body or the carpet. The shed hair that is doubtful as to its impurity is considered pure, because the original state is purity. We advise turning away from obsessive thoughts.