Is it necessary to wash the prayer rug, trousers, and everything touched by a cat that had traces of feces on its rear and tail, and does the cat become pure by the removal of the traces of feces, or must it be washed with water?

We advise you to strive against obsessive thoughts and disregard them. Dry impurity does not transfer from a dry impure object to a dry pure object. When in doubt about the transfer of impurity or the impurity of something, the default is that it has not transferred and purity remains, or that the object is pure. Do not rule something impure except with absolute certainty. It is necessary to wash an impure object with water to purify it, except for the ground and what is connected to it, for which drying by sun and wind is sufficient for purification according to many scholars.