Must the hand be washed every time after cleansing with water (istinja') following urination, or only when in doubt? Is it sufficient to pour water only to wash away impurity from the hand or body without rubbing? And must the flowing water from the impure area and the area it flowed onto be washed off?

There is no obligation to wash the hand after Istinja (definition) (cleaning oneself after urinating or defecating) unless it has been contaminated with urine and water has not flowed over it. The default is that there is no contamination. To purify a najasah (definition) (impurity) from the hand or anything else, one must first remove the physical substance of the impurity, then pour water over its location and increase the amount until it covers the area and removes its color, taste, and smell. Rubbing, squeezing, or the water separating are not conditions.