The animal imagery used by Bronte in Wuthering Heights offers more for the motif of the influence of the forces of a nature upon the plot as well as an insight into the characters. For instance, Heathcliff, who is described in wolfish terms by Catherine, describes those he disparages in animal terms. He views Edgar Linton as a "lamb" who threatens "like a bull," and he calls Linton, his son, a "pulling chicken." He even describes himself as he dies, "...the beast has changed into carrion."
You might want to compare the various animal descriptions that occur in this excellent novel to the depiction of Heathcliff and how Bronte often goes out of her way to present him as a strange mix of human and beast. One of the many Gothic aspects of this classic is the way in which Heathcliff is never presented simply as a human, and often the descriptions we are given of him challenge any easy assumptions we can make about his humanity. At various points he is described as being like a wild beast in his savagery, and the inclusion of animals helps stress this ambivalence about his character.
I see that Heathcliff is ferocious. He lived on the streets and was adopted, but was never accepted by his "adoptive brother." He has always felt like an outsider in that house, and when his "father" dies, he is treated badly by his brother, Hindley. He is treated almost like a servant, perhaps feeling as if he is being treated like an animal, and perhaps his aggressive nature is necessary—in his mind—to survive.
When Heathcliff overhears Catherine saying that she loves Heathcliff but will marry Edgar, his heart is hardened and he runs away. While he is away, he will need to reinvent himself, but he has lost Catherine and instinctively, Heathcliff will become more aggressive.
Perhaps all of this simply signifies Bronte's belief that there is a fine line between the sophisitication of a "civilized" person, and the animal-driven instincts that people rely on to survive or win.
You are noticing evidence of a motif in the novel. There is a clear animal motif -- there are actual animals, symbolic animals and animals referenced metaphorically. You and the above posts have some good ideas in regards to what these animal references mean. There isn't one answer, necessarily, but what you need to do is brainstorm what qualities you associate with the animals that are mentioned, especially as it relates to the context of the reference. What do you normally associate with dogs? Does that work in relation to the character that is being compared to a dog? or the the situation? is the reference actually used ironically to suggest the opposite? Once you collect your examples and ask yourself these questions, you will be on your way to an interesting analysis of Bronte's writing in regards to this topic.
Wuthering Heights is an example of Romanticism in literature, the primary literary movement of the 1800s, both in Britain and the United States. Heathcliff's wild, untamed nature contributes to his development as a Romantic hero. He feels deeply, but there is an element of danger in him, too. Heathcliff is an example of the Byronic hero in Romantic literature, modeled on Lord Byron's central character in Childe Harold's Pilgrimage. Examples of the Byronic hero appear throughout literature, from Lord Byron's time to the present day. Another element in Romantic literature is nature. The many references to nature in Wuthering Heights reflect the influence of Romanticism in the development of the novel.
I think the object is to stress the uncivilized versus the civilized, a theme throughout the story. We can pay attention to which animals are mentioned with which characters, and how they are described in connection with which family or character.
Animal reference are used to Heathcliff a more beast like aurora or personality. One example I can think of is when the dog that is raised by Heathcliff is described as a the "devil." When you think about it Heathcliff was the one to raise the dog to be what he and only evil can only teach evil. Also there is the time when Mr. Earnshaw brought Heathcliff in as apart of the family ad everyone instantly see him as a savage and a beast and when Mr. Earnshaw died Hindley started to treat him exactly like one. So Heathcliff has been doomed to be seen as anything but an animal since he walked into the Earnshaw's home. I like to fancy that Heathcliff is the Beast without a beauty and he continues to stay this way because his true love is no longer in the realm of the living.