Does everything my hand touches become impure because I suspected it was impure after touching my child's clothes, which were wet with urine, and then washing my hand?

The hand becomes impure by touching a place of impurity wet with urine. As for the transfer of impurity from the hand to something else, if the touch occurs after washing the hand, there is no impurity. If it occurs before, impurity is not transferred unless there is certainty that one of the two touching bodies is wet. Indeed, some scholars believe that it is not transferred in this situation at all. And according to the view that it is transferred, you are not obligated to purify what you doubt has been touched by the impurity, but only what you are certain has been touched; for the default is non-transfer.