Water moccasins, or cottonmouths in the South are common examples of snakes that can bite underwater.
Yes, snakes breathe through a small tube at the bottom of their mouths and can close it when underwater enabling them to bite without drowning.
There are lots of snakes who are fully water snakes they have gills they never com up for air so thats a yes
Yes. There are about 55 species of true sea snakes. Sea snakes can remain submerged for several hours, possibly as much as eight or more.