What is the ruling on touching a handkerchief of unknown purity in a wastebasket with a wet hand, and what are the implications of that?

Nothing is incumbent upon you. The original state of a handkerchief about which you doubt its impurity is that it is pure. A thing is not judged to be impure merely by doubt. Rather, the original state of purity is maintained until certain and firm conviction of its removal is attained.