^ He's more of a dog person, and cats are trying to take over the world? 

All jokes aside, I think it's trying to portray him as the type of guy who seeks revenge? 
the dog had an audience of people who saw its rescue (where as the girl alone was caring for the cat), so I assume he left the dog alone because its death may have aroused the curiosity of one too many people than he'd like. portraying him as a character who takes revenge makes sense, too, & as the show goes on I see him as showing more of his villainous qualities. in which case: the cat came after the dog. idk if it's something the writer even thought about too much tbh, because it does seem minor in comparison to the bigger picture. still, killing the cat that way feels like it was meant to reveal more about what kind of character he is in addition to starting a chain of events. if he didn't kill the cat then he wouldn't have needed to kill the girl, & if he hadn't killed the girl then there'd be no suspicious detective sniffing around him. I'm now three eps behind so idk where the drama wants to go with this plot point or what new info's popped up, but that's my guess for the time being.
You need to catch up to ep 11 @artwit. Things revealed about the vet left, right and centre. I'm just amazed, he spent 5 years going around, being so careless with leaving dead bodies behind, and not even bothering to check if Hyun Ji is properly dead or alive, by going through the hospital rotas. From what I see so far, though, he doesn't kill everybody, just anybody who he perceives will cause him harm. It's almost like a little kid lashing out, and plus what evil spirit cares what a human mother thinks? I'm not entirely convinced he's 100% possessed, there might just be a human soul corrupted by pure evil. His motives still remain the most interesting though and dying to know what THING Hyun Ji was meant to have in her backpack, and Park Ji Hoon had....also why kill the peeps, and ask questions later? They're in no position to answer are they?