Does that which the shoe touches become impure if the bathroom floor had been splattered with drops of menstrual blood, and water was poured over it without removing the blood, and the shoe was damp? And what is the ruling if it is impossible to wash something that has become impure, such as a carpet?

If the color and odor of blood are removed from the bathroom floor by pouring water, then the floor and whatever part of the shoe touched it become pure. The carpet will not become impure from the wetness of the shoe. However, if the water does not remove the color or odor of the blood, then the impurity remains, and whatever part of the carpet was touched by the wetness of the shoe must be washed. The carpet does not become impure based on doubt about it being affected by impurity.