Perhaps not an addiction, but a main character is forced to be constantly supplied with Eridium, and when cut off (necessary to progress story) dies due to their dependency. Another main character could be seen as being addicted/dependent on Eridium as well.
The level Wildlife Exploitation Preserve is a facility where creatures are experimented on, one of which is a main character's pet. A creature is dragged behind a truck in the opening cutscene.
Alien animals are a common enemy, but there is also mention of Brick's dog being killed as well as a mission that ends with Mordecai's bird Bloodwing dying
Not as intense as the first game, but enemies can still be eviscerated in numerous different ways. Killing an enemy with a head shot will cause large chunks of flesh and blood to fly in every direction.
One Character's backstory is revealed through audio logs, in which you can hear their parents telling them to run before it is implied they are killed.
In the Tina DLC it is mentioned that the queen is kidnapped but the context of the DLC is a D&D style game.
In the Son of Crawmerax DLC Sir Hammerlock is kidnapped.
a playable downloadable character named Krieg the Psycho has an optional ability that will cause himself to violently strike his own head randomly to quiet the voice/s he hears that act like a conscience
There's an entire enemy group called "psychos" who babble nonsense while trying to axe-murder you or blow you up. There's also a player character who has Multiple-Personality Disorder, who's one of these guys. He has an entire skill tree devoted to detailing how useful his mental illness is in combat, and has an inner monologue about how he hates his surface personality.
Face McShooty asks for help in assisting his suicide. It's played for laughs, however. There are also enemies called "Suicide Psychos" who try to blow you up by priming a grenade and charging at you.
Not necessarily PTSD, but Tiny Tina was driven insane by the loss of her parents, and in the Dragonkeep DLC is shown to be struggling to cope with the death of a friend.
One of the most common type of ennemies are erratic men referred to as "Psychos", and most human ennemy type have a small variant refered to as "Midgets"
The weapon "the nukem" has a mushroom cloud as its explosion which is reminiscent of nukes, but no actual nuclear weapons are used in the game at any point