What is the ruling on water splattering onto the body after washing a contaminated hand or the private parts after urinating? Is it considered pure or impure? And what is separated water?

The water that separates after purifying an impurity is pure, and you are not required to wash whatever of it touches you. It is sufficient for istinja' (cleansing) from urine to wash the area with water until the impurity is removed, and whatever splatters from it after its removal is pure and does not require washing.