What is the ruling on places that have been walked upon after touching an impurity, given the difficulty of washing the entire فرش (bedding/carpet)?

Unless you are certain that the defiled garment has contaminated the ground, the original presumption is that the ground is not defiled, because the transmission of impurity is not determined by mere doubt. If you are certain that the ground is defiled, then there is a difference of opinion regarding the transmission of impurity from it to the foot of a walker who steps on it. If the matter becomes difficult, there is no harm in adopting the easier opinion. One must beware of obsessive doubts and succumbing to them.