It's possible to abandon rats, but those were already wild to begin with and you can only pick them up for your inventory and for rat races, not really as pets.
There's a scene of a mob burning a woman alive. It's not for the spectacle of the audience, as the depiction isn't very focused nor explicit, but the woman is killed in front of a crowd.
The Bachelor is implied to be gaslit by The Powers That Be, the Inquisitor, and possibly the Kains, especially near the end of the game. However, it is only implied. There are constant examples of all of the main characters being manipulated and lied to, but not under the intent of convincing them that they are crazy (which is what gaslighting is).
Rats are a common enemy in the game, and thus can be killed. No gore is shown, the rat just falls on its side.
Bulls are sacrificed for a ritual, though this is not depicted on-screen.
Yes, but it is not directly portrayed and can be missed. Here is any mention or implication I can remember.
1. Though easy to miss in the game, one of the dancers in the Broken Heart pub is a minor.
2. Through his quest on day 5 in the Bachelor Route, it is easy to interpret the usurer Var as being predatory to his adoptive daughter Willow.
3. A missable conversation with Andrey in the Changeling Route (I don't remember the day, my apologies—I think it's near the middle of the game) brings up the possibility of Clara, also a minor, dancing in his pub.
4. In the Haruspex route, I believe one or two female NPCs (Eva and Lara, I think?) will express fear of you, implying fear of SA.
5. One of the Dogheads in the Stillwater in the Haruspex route references townsfolk keeping their children from interacting with the Kin out of (racist) fear of SA.
If you speak to Eva again on Day 2 of the Bachelor's route after the conclusion of the "Let's run!" sidequest, a line of dialogue by Eva expresses intent to commit suicide this way, and an available response by the player suggests hypothetically lending his assistance.
Straight razors can be found (e.g., in garbage bins) and either traded or sold, but are not otherwise utilized. Knives are wielded as weapons by the most common enemy type in the game, and possibly by the player. The player can also wield a scalpel this way.
On Day 1, a woman is burned at the stake by a group of townsfolk. This is depicted on-screen, and her screams can be heard. This can be encountered in any route.
On Day 10 of the Bachelor route, a character attempts suicide by self-immolation. This attempt is not successful, and the character is unharmed.
Several people are hanged in a cutscene on the beginning of Day 7. This happens in every route regardless of the player character that's chosen, or choices made during a quest. The bodies are depicted on-screen at the very end of the cutscene, only the aftermath is seen and not the process of hanging. The bodies are not present anywhere in the game, only in this cutscene.
Several characters make reference to being hanged by The Powers That Be, should they return to the Capital without achieving their goal. This is never depicted on-screen.
A minor plot-point has to do with an off-screen group kidnapping children, and a character is revealed to most likely have been a part of this group, having abducted a baby.
Several characters have dead parent(s) and the town has a many children with no parents present. The Haraspux's route references his dead father many times.
There are no literal ghosts, but the plague clouds resemble ghosts and appear randomly / chase you. Two characters are able to speak to the dead and multiple characters mention seeing ghosts.
The police and military are shown to be very flawed and complex. The Changeling Route has in-universe copaganda that she believes in and champions for, though. Meanwhile, the Bachelor and Haruspex routes involve a lot of soldier and/or police murder and disdain.
Yes, but not in "oooh mentally ill people are scary" way. Many characters are violent, some of which are mentally ill. It's quite nuanced and not portrayed in an ableist way.
One of the player characters receives a suicide note from one of their friends, to later find that they have died. Later in the same route, the player must talk another friend character out of their on-screen suicide attempt.
Three characters mention wanting to commit suicide. Two of them write a suicidal note which is sent to you. (Both on Day 07 of Haruspex / Bachelor routes)
In addition to the aforementioned character death, on Day 10 of the Bachelor's route, another character attempts suicide by self-immolation. This attempt is not successful and he survives the rest of the game.
As well as this, several characters will express suicidal thoughts or ideation when under stress.
Self-sacrifice through giving up one's own life is a reoccuring theme throughout the game.
The treatment of the usurer, both from a writing standpoint and in-universe, is questionable. (He is referred to as "the Hunchback", but from what I remember it doesn't get much beyond that.)
An LGBT character commits suicide, though it is unrelated to her sexuality. Another LGBT NPC has the possibility to die, though it is not a scripted death.
No explicit sexual content is present in the game, but several characters can flirt/make passes at one another in a suggestive fashion, or have dialogue that can be interpreted as sexual.
Several women are scantily clad in-game.
Notkin and his gang are homeless children, other homeless children are both mentioned and shown (such as one that lives in an abandoned building), the Haruspex is briefly without a home/shelter and the Changelings parents permanently kick her out of their home on a certain day.