Dogs can be killed, but it is not forced in game.

There are no cats in Skyrim that resemble housecats, but there are giant sabrecats that are often hunted by NPCs but they don't resemble cats very much other than their paws and body structure.

Horses are common in Skyrim and are almost always killable. (You can kill Shadowmere but she respawns at her death place after a while) and in Helgen, a dragon is shown to kill a horse in the middle of the town.

There are many other animals in Skyrim such as chickens, wolves, foxes, hawks, and etc. Some of them are programmed to be killed in certain events, and some NPCs will be seen hunting them.

Frostbite spiders, charus, charus hunters, are all enemies

The character named Ceciro is a murderous Jester who is very clown-like and very creepy.

There are lots of ghosts found in Skyrim, as NPCs in quests are random encounters in dungeons. There's even children spirits in some quests.

Many enemies can easily sneak up on the player, such as Frostbite Spiders that can descend down from the ceiling unnoticed until they leap at you.

Frostbite Spiders are a common enemy which are just giant, venomous spiders.

In Skyrim, dragons are a common enemy, and you are forced to kill many of them such as Alduin, the main antagonist.

If you follow the Dark Brotherhood questline, there is a head-squashing.

As other comments show, there are storyline child deaths. In gameplay outside of quests however, children cannot be harmed outside of Player-made modifications to the game.

No character is explicitly called out as LGBT, but the player character and some NPCs (revealed to be LGB by being same-gender romance options) can die during gameplay.

There are many drunk NPCs in Skyrim that spend their septims on ale and mead, and often ask the player for some money.

I used to be an adventurer like you, then I took an arrow in the knee.

An orc will stand on the roadside, awaiting a good death (IE, in combat with a powerful monster or warrior). You can offer to be his good death if you so choose.

Underneath Dragonsreach, there is the skeleton of an unknown person, floating in the water. The player can drown if they are underwater for a long period of time.

Due to glitches, some NPCs can be sent flying in the air from being attacked by a giant, and then fall back down, dead.

If you join the companions, a werewolf has it's blood drawn out by having someone else slice it's wrist open. Not pretty considering it fills a bowl with blood.

Skooma is a common drug in Skyrim, and there are many quests involving skooma addicts. In the Redwater Den added by the Dawnguard DLC, there is a dead khajiit bandit who died from skooma overdose.

The afflicted are a group of mortals cursed/blessed by the Daedric Prince of disease, Peryite. They have the ability to projectile vomit poison on opponents.

In Riften, there is an orphanage in which there is an old lady seen abusing children verbally, and implied to abuse them physically too.

In some places, you can find burnt corpses. (Ex. In Helgen some burnt corpses are found, burnt to death by a dragon.)

In a special dungeon, there are thralls kept by a man whose will is trapped inside of soul gems. If the soul gem is removed from the pedestal, the thralls will turn on their master and attempt to kill him for imprisoning them.

The Thalmor can be found torturing Stormcloaks in their embassy, and in the Dark Brotherhood questline, you have the chance to torture some people until they give you money.

In many dungeons, it's very crowded or small. And if you get arrested in Skyrim, many jail cells are very small. (Especially the one in Windhelm.)

Some places of worship are populated with comatose sick people being tended to by priests, but the resemblance to a hospital ends there as they lack any modern technology.

A few characters are addicted to alcohol, and a few are addicted to the drug Skooma - one optional side quest has you retrieving some Skooma for an addict who is experiencing withdrawal symptoms.

When fighting, there is alot of blood, and you can even decapitate your opponents, and in some quests, you can harvest human flesh and human hearts.

The player has the ability to attack anyone in the game at any time.

Optional, but in the character creation screen, the player can choose to give their character scars over their eyes, which could imply mutilation depending on the player's personal lore for the character.

Bandits may often yell at you depending on your race. (Ex. 'You'll make a fine rug, cat! if you're a khajiit) (Ex. "Die, you Orc filth!" if you're an orc)

Never anything explicit, but it is talked about occasionally.

When opening Skyhaven Temple, the player cuts their hand with a dagger and uses the blood to open the door. This is all shown in an animation.