Must everything touched by a cat whose paw stepped on an impurity and then roamed the house be purified, knowing that this is very difficult, and is there a waswas (whispering) concerning the impurity of the remote control?

The treatment for obsessive thoughts is to ignore them and pay them no mind. The preponderant opinion is that the urine and dung of animals whose meat is permissible to eat are pure. Horses are among the animals whose meat is permissible to eat according to the majority of scholars. Therefore, the paw of a cat does not become impure by stepping in a garden containing horse dung, and there is no harm in it walking inside the house. If you doubt the impurity of the cat's paw, the original state is purity, and it is not obligatory to purify something whose impurity is doubted. As for purifying the ground and carpets, it is only necessary for prayer upon them. The obsessive thoughts regarding the impurity of the remote control should be ignored.