Is the rug that Paul, my cousin's daughter, stepped on considered pure, and what is the ruling on touching things after touching the rug with wet hands?

If you are certain that the wet impurity on the child's shoe touched the carpet, it is obligatory to purify the carpet by pouring water on the spot of the impurity. It cannot be purified by the sun or wind. However, if you are not certain that the shoe became impure, or if the impurity was dry, then the carpet remains pure, because impurity does not transfer between two dry objects. The fundamental principle is the persistence of purity unless impurity is certain. One must avoid obsessive doubts (waswasa).