There is no 'rule' about who can get killed in a horror movie.
The Exorcist
Yes, it would most likely be classified as horror or suspense.
I had been looking for the name of this movie for a long time. I thought it was a horror movie but I have found out that it was a comedy called "Arnold" and came out in 1973.
Horror is most popular
rambo
the exorcist
Horror
the shining
The Exorcist (1973)
For the most part, the quintessential horror film with a dog in it was the film "Cujo".
Fragile (2005) starring Calista Flockhart