How can impurity be washed and purified from different surfaces, porous materials, and clothes, taking into account the quantity of water and the number of washes? And are modern washing machines sufficient for that? And what is the ruling on using detergents that may contain animal products?

Every solid, smooth surface is purified by wiping it and removing the visible impurity. The number of washes is not a condition for purifying an impurity, except for the impurity of a dog. As for carpets and rugs, they are purified by pouring water on the impure spot and then extracting it. The water used to remove the impurity should be abundant enough to remove the impurity. If an impurity falls into water, the water does not become impure unless its color or smell changes. If an impure garment is washed in a washing machine and the trace of the impurity is removed, then it becomes pure by that. There is no harm in using cleaning agents of animal origin that were originally impure, if they have transformed into another substance that differs from the impure entity itself.