Does the hand become impure and require washing when an insect generated in impurity is killed on it? And what is the criterion for insects generated in impurity: does it include any place in the bathroom, or is it restricted to the place of defecation itself?

As long as the questioner follows the Hanbali school regarding the impurity of what is generated from impurity, then this generated thing is impure, whether alive or dead. If it falls on the hand or garment while wet, the place becomes impure. If it is dry, it does not cause impurity. Ibn Qudamah said: "What is generated from impurities, such as toilet worms and cockroaches, is impure, whether alive or dead." As for what is on the wall or the ground and was not generated from impurity, it is pure. If you doubt whether it is pure or impure, the default ruling is purity.