Answer:
I am an animal named after the animal that I eat.
Those with eyes bigger than their stomach will definitely leave the restaurant with one of them. It is named after a house pet as well.
The fans of the pied piper's smooth sound who are feared by elephants
A blackbird similar to, but much bigger than a crow
Who is it that rows quickly with four oars but never comes out under his own roof?
A man goes out in heavy rain with nothing to protect him from it. His hair doesn't get wet. How does he do that?