No, this isn't Wizard of Oz, but animals populate our writing every bit as much as people do. I like to have horses in much of my writing, but I also have a story where shape-shifters are the dominate population and any mammal of suitable weight was possible, though a werwolf is my main. I even have one story where human DNA and wolf DNA were mixed with very unexpected results.
If you're going to have animals in your stories, remember that they have a certain way of acting and reacting to certain stimulus. Even those half animals or wer-animals should be part animal in their actions and reactions. I try for a wolf feel in my half wolf character, though in nearly every way he looks human. My werwolf has a wolfish temper when it shows up.
Another thing to remember is when your people encounter an animal, they too will react in a very predictable manner. For instance, if a man encounters a bear, what will he do? What will the bear do? There are several options depending on how the encounter started. I read a story not long ago where a woman was tackled by a lion. The lion was very human by day but very lion (with a human mind) by night, but there was no way the woman would know that at that moment. The moment was handled very poorly in the story. Before she fainted, she took the time to notice how kind the eyes looked. I'm sorry, if I were tackled by a lion, the last thing I would be noticing would be how kind its eyes were; I'd be far too worried about where those huge teeth were going and how to keep them away from their target.
I grew up with horses and I like to think I am familiar with most kinds of temperaments. Just like your people, each member of a herd of horses would have its own personality. There is the ever-watchful stallion and the wise lead mare, the playful colts and the new-born just learning about his feet. There's the mare who bites everything that comes close and the gentle one who tries to mother everyone.
When you're writing a scene that involves animals, even in the background like your person's cat, remember them. Don't announce the cat and then forget it. Cats may not do much more than sleep, but they seem to manage to sleep right there in the way most of the time. The family dog will always greet whoever comes to the door in some manner, and if there's two dogs, they may greet differently.
What animals do you include in your stories? What is your favorite scene involving an animal, half animal, or wer-animal?