Does the فرش become impure if a child is wiped with wet wipes, then the spot of impurity comes into contact with the فرش and no color or smell of the impurity appears? And does the impurity transfer from it when one sits on it? And what is the ruling on excessive doubt concerning impurity?

Removing impurity with wet wipes is insufficient for purification; rather, it must be washed with water. If the wipes used to remove the impurity were wet, they become impure and would defile anything they touch. However, if they dry, they do not defile anything dry they come into contact with.

After the physical impurity is removed, but its legal status remains, if the impure spot is wet and touches something dry, scholars have differed: The Malikis and Hanafis hold that it does not become impure, while the Shafi'is and Hanbalis hold that it does become impure if it is wet (not merely damp).

Obsessive doubt concerning impurity and doubting its presence is a type of waswas (definition) (whispers from Satan), and one must cast it aside and pay no attention to it, because the default status of things is purity, and certainty is not removed by doubt.