What should be done about a cat's repeated urination on the floor and carpets, and are the carpets considered ritually impure, rendering prayer on them impermissible, and are the feelings experienced by the questioner mere whisperings [from Satan]?

Unless the impurity of the cat is ascertained, the original state is its purity. Impurity is not judged by mere doubt. If it is possible that what is on its body is urine or something else, the original state is purity. A cat is not judged to be impure, nor is impurity transferred from it, unless this is ascertained with absolute certainty.