What is the ruling on washing carpets with water and soap without pouring water on the spot where urination occurred, and does it become pure thereby, and is the house considered ritually impure as a result of walking on the water that separated from the carpet washing?

Filth is not purified from carpets merely by drying. Whatever comes into contact with the impure spot becomes impure if either of the two surfaces is wet. A carpet is purified by washing it with water and using an abundance of it. The wash water is considered impure if it separates from the place while it (the place) is still impure. However, if you doubt the state of the wash water, the default is purity. Past prayers performed while ignorant of the impurity do not need to be repeated. The default state of things and places is purity, and this default is not changed merely by doubt or obsessive thoughts.