Considering that many crocodilians eat fish and they drag their prey into water before eating it, I'd say yes, they can bite underwater.
Alligators aren't the only landlocked reptiles known to swim many miles at sea. Komodo dragons are known to swim from island to island and eastern diamondback rattlesnakes have been found as many as 30 or more miles out from shore. Human breathing rests at the bottom of the breath, when the person is least buoyant. Rattlesnakes, on the other hand, rest with their lungs filled to capacity, meaning that they can effortlessly stay afloat for a long time. While we're on the subject, Crocodiles and alligators have special lungs called multicameral lungs that don't work like ours. Their have many chambers that spiral around each other branching off from one main chamber. I don't know what this means for resting period of a crocodilian breath.