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.
In the Phobos level opening scene, Doomguy approaches a seated personnel member, who has a keycard he needs to get through a door. The worker hunches back in his seat as the distance between the two steadily decreases.