Are impure clothes purified by placing them in the washing machine with pure clothes? Do pure clothes not become impure by contact with impure ones? Likewise, does the hand not become impure by touching impure clothes while they are wet?

Washing with purifying water after soap, or before it, removes impurity, whether the washed item comes into contact with the water or vice-versa. If the physical trace of the impurity is removed by water and soap, while its legal ruling remains, then it does not transfer to anything wet or dry.