Not during the events of the game, but the protagonist once had a pet rabbit that was killed by demons, and you can find an empty cage with a rabbit's foot in it in the hub.
While this doesn't happen during the in-game story, it is known that Doomguy had a pet bunny which was killed during one of the invasions. This bunny is seen in each level as an easter egg (which may be interpreted as a hallucination), and an empty cage in Doomguy's base.
Certain finishing moves involve stabbing enemies through the mouth, decapitations, or boppin them on the head hard enough to smash their heads into their chests.
In one level, the player traverses a factory in Hell where human souls are barbarically tortured in order to extract energy and sometimes turn them into demons.