Why are those snakes called cottonmouths?

No, it is not. It is poisonous.

No, but both are in the pit viper subfamily, Crotalinae.
No, however it is in the same family (pit viper, meaning they have "holes" on both sides of the face between there eyes and nostril (these are where they get the term "pit" in pit viper, the viper part comes from the way there fangs are designed, they fold back in the mouth and "spring" open and forward when they are in the act of biting) . These snakes are also cousins of the copperhead.