If a family member washes his clothes, and those clothes still have ritual impurity (najāsa) after being put in the dryer, must they be cleaned and are they considered impure? And must the washing machine be run to rinse them to purify them from the adhering impurity, especially since the washing machine fills with water and drains it twice? What is the ruling in the case of unknown colored spots, or blood spots on clothes after they have been washed in the washing machine?

You must ignore obsessive thoughts and not succumb to them. To wash away impurities, it suffices to pour water over the impure spot until it is submerged and the water separates from it without changing [in color or smell], and wringing is not necessary. You are not required to wash the place where your family washed clothes if you doubt their purity, nor is it obligatory to rinse and agitate the washing machine before using it if you doubt its purity. Likewise, you are not required to wash a garment in which you find traces you doubt are impure, because that is an effect of obsessive thoughts.