Yes they do. It is so they are ready for summer. Or the other seasons.
Alligators can slow down their metabolism during cold spells and live off their body fat, but this isn't true hibernation.
Not all animals do but a lot do like a bear.
Yes, Most eel do hibernate in the winter. The most common eel (That hibernates in winter) is the Grizz Eel. It is more efficient at catching prey (And getting nukes) than all other eels, Even better...
Most salamanders hibernate during winter inside rotting or decaying logs, under leaves, or underground.