The protagonist is shown to be in a sour domestic relationship with his wife and shoots himself in the head at the ending of the original game because of her nagging, but nothing is visually shown.
While the dogs were modeled and textured after Vince Desi's real dog Champ, the gore textures appear to be the same
ones used for human characters. And the real Champ is dead, he died in 2011. So I doubt Desi harmed Champ for any assets in game.
In the beginning scene, the main character, Postal Dude, kicks his dog in the stomach after seeing it had peed on the ground. It whimpers and runs away but is relatively fine after.
You can also use cats as gun suppressors, but it is an optional thing and not forced upon the player.
Though there are implications of in-game torture, nothing is explicitly shown. The optional harassing of NPCs can be considered torture at their most extreme.
Vince Desiderio made it a point in the first Postal game that he refuses to kill off any children in his videogames, and that applies to Postal 2 as well. No children appear in-game.
In the DLC the main character has a voice in his head which can sometimes speak for him and to him, but does not control his actions and they have different voice actors.
Despite the no's, it is implied that the protagonist shot himself at the end of Friday. It is not shown on screen however, and the weekday cutscene blacks out with the gunshot. You can also harm yourself (the protagonist, specifically) in-game with cacti, falling down from a tall height, lighting yourself on fire (and not extinguishing yourself), etc.
It's never called D.I.D. but the DLC includes Postal 3 Dude as an alternate personality/voice in the Postal 2 Dude's head. He is called Alter Dude in the subtitles. He speaks for the Postal Dude a few times and can sometimes taunt him or the player by commenting on their actions in game or giving them hints. [DLC SPOILERS...] He eventually escapes from P2 Dude's head and has his own physical form and then he disappears from reality at the end.
There is a button that can be pressed at any moment to make the player character commit suicide via grenade. There is an option to undo called "wuss out." The game ends with an off-screen attempted suicide, only the gunshot is heard.
There are moments in both DLCs where a possibly disorienting moving filter is overlaid on the screen. They are more frequent in Paradise Lost as it appears whenever Alter Dude speaks in Postal Dude’s head.
If you quick save the Postal Dude will make a remark, it also might happen if you are arrested eg ("It's not my fault, book the kid with the keyboard")