Dogs don't seem to feature prominently, but in their place there's plenty of cat fighting to serve the same purpose. There is a character that trains cats to fight and asks you to send him cats with "combat experience" who can "take a punch."
Maybe not in a way that will be actually effectively horrifying, given the artstyle, but the game is about breeding cats to the point that they mutate, and encountering mutated/disfigured creatures in the wild.
There are cultists who cut themselves, and it is crudely animated. There are abilties that allow you to hurt yourself to gain a gameplay benefit, one is called "self harm." Not sure if this will be triggering to anyone because of how cartoonish and detached the portrayals are.
Steven can describe your cat as "queen of the short bus" after inflicting your cat with a fictionalized memory disability as punishment for save scumming.
Autism is framed as "-2 Charisma, +2 Intelligence. The spells you were born with cost 2 less mana. Your other spells cost 1 more mana." This is an oversimplification but the game is not trying to depict each disorder/condition in it accurately, isn't trying to be harmful, and is not meant to be taken entirely seriously.
Various vulgar language as well as references to crass subject material, on-screen and off. Steven mentions the game he's originally from which itself has a vulgar title.
Characters in the game treat cats as disposable resources, make them fight, use them for science experiments, etc. The tone is satirical and most of the violence tends to be the cats actually fighting each other.