Is the prayer of a woman valid after her sock, with which she prayed, touched water that she used to wash a tiny drop of blood from a wound, and what is the ruling on her prayer if she knew her sock was contaminated with this water?

The water that separates after washing an impurity, if it has not changed color, smell, or taste due to the impurity, is considered pure according to the preferred opinion. Therefore, the sock you walked in over this water will not become impure, your prayer with it is valid, and you do not need to repeat it. One must beware of obsessive whispering (waswasah) and turn away from it.