If you have trouble distinguishing between seemingly synonymous words, Google Images will be happy to lend a helping hand.
Coop:https://www.google.com.br/search?q=coop ... h&imgdii=_
Cage (please ignore Nicolas Cage's pictures)https://www.google.com.br/search?q=coop ... h&imgdii=_
Hutch:https://www.google.com.br/search?q=hutc ... h&tbm=isch
A cage is a "jaula" or a "gaiola" and can be used for a number of different animals, from birds (as is the case of birdcages) to bears. A "coop" is usually a "galinheiro". I don't like going into elevators, because it makes me feel "cooped up", just like all those hens. A "hutch" is a structure where rabbits and other similar animals can be kept. A hutch can also be a piece of furniture similar to a "cristaleira", as you may have seen in the pictures.