In the manga, not in the tv show

(MANGA SPOILERS) As a child, Himiko Toga killed a bird due to her fascination with blood.

No animal abuse occurs on screen. However, Nezu (a sentient mouse with a Quirk and the principal of the school) was reportedly experimented on in the past, and it caused him to develop a sadistic side (which we see during his fight with Mina and Denki).

No dogs are seen fighting.

One episode involves Present Mic getting assaulted by a group of bugs travelling from underground, and they end up covering his entire body. It's not extremely important, so skipping this particular scene is optional.

Yes, in Heros: Rising the character Nine is able to conjure dragons that get killed.

There is a very minor character with snake hair.

Smoking but that’s about it

In season 4 and in the manga, a drug called "trigger" is produced. It's used to amplify people's powers.

Yes, a villain in a Yakuza group abuses alcohol.

No cutting themself, but the Hero Killer Stain uses knifes to cut other people quite a bit.

Yes, the character Mr. Compress loses his arm on-screen and the character Tomura Shigaraki loses fingers.

One of the villains gets his fingers cut off off screen

The monster creatures known as Nomu have been burned to death but flame hero endeavor

In season 3, a villain named Moonfish is obsessed with consuming flesh, though he's never seen doing so onscreen.

Aizawa is pressed to the ground by a strong enemy holding his head down. He suffers injuries and is ultimately fine, but at the time there is the threat of head squashing.

One of the villains uses his teeth as weapons. His name is Moonfish. They are used to fight and break like knives.

Episode 17 of season 2 (SPOILER FROM HERE ON) As a nomu is flying away with a captive, Endeavor launches a spear made of fire that strikes the creature directly in the eye. The actual blow is hard to see unless carefully watched due to explosions, but later in the episode after the major battle with Stain is over it reappears. Once it shows back up it has a bleeding empty socket on one side of his head until the creature is gone.

One of the main characters frequently injuries his fingers, which are shown to be various shades of purple and bent out of shape, causing him pain.

Midoriya Izuku is constantly breaking his bones due to his Quirk. It is a large plot point. Other characters are also frequently getting harmed.

A 5 year old boy named Kota lost his parents when they fought against a particularly strong villain.

It's a big plot point in season 3

Shigaraki’s sister is killed in the manga

Toya Setsuno was cheated on by his fiance.

endeavor exists

There is a scene in a hot spring and o e in a female locker room. They aren't that sexulised, though.

There are some flashbacks to a deceased character, but I wouldn't call them a ghost at all.

There are elements of mind control, but no one is possessed per say

There is an ongoing/common gag where Ochaco laughs and ends up spitting.

All Might vomits blood as a reoccurring joke

Todoroki's mother is kept in the hospital by her abusive husband for mental reasons, but I believe it's a regular hospital, not a mental institution

Kaminari Denki does use his Quirk to electrocute people though

The protagonist is frequently sent to the hospital due to his tendency to break his own bones using his quirk.

Himiko Toga uses a syringe while she fights.

Todoroki's abusive father tries to convince him that his mother, whom he drove to madness, was a bad person, and he's protecting him by sending her to the mental ward, even though he's the reason she went mad in the first place.

y'all PLEASE. body dysmorphia and body horror are NOT the same thing

Several characters’ quirks are dependent on their calorie intake and there are scenes showing them binge eating for that purpose. It’s not a mental health issue for those characters, but watching them eat like that could be triggering for certain people.

There are some scenes that greatly resemble anxiety attacks.

During the Yakuza ark one of the villains controls the ceiling, walls and floor of a building. I have personally felt this scene to be quite claustrophobic.

No, but theres a scene in the first episode where the character Katsuki Bakugou tells the main character Izuku Midoriya to kill himself

One of the main villains has a tic that results in him scratching his neck. The more upset he gets, the more he scratches. There are a couple shots that show red lines, depicting that he's hurt himself. Though not cutting in the usual way, I'd definitely still be wary.

Yes and no, because according to Japanese standards for anime, all strobe-like effects that have the potential to cause seizure already have a seizure filter over them.

Occasionally used as first person POV to show a character looking around during a high tension scene.

There is a brief repeated detail of the first baby being born with a Quirk, but it doesn't show any actual childbirth, you just see the baby in the hospital

Minoru Mineta regularly sexually harasses his female classmates-he makes inappropriate comments about them, tries to peep on them whenever possible, and has groped them more than once. Unsurprisingly, a LOT of people hate him, and what doesn't help is that it's played for laughs and he hasn't been expelled for his behavior (yet).

There are no actual sex scenes or anything like that, but I'm going to count fan-service as sexual content

A villain misgenders a trans female character, though her allies do correct the villain in question.

The character Magne, a trans women, is killed in a violent way. The villain Overhaul uses his quirk on her, which makes her explode in a bloody way.

In the manga there is a group that is compared to the KKK and discriminates against people with mutant quirks.

As of now no black characters have died in the series.

In the 4th Season, Red Riot makes His Work study with "The BMI Hero: Fat Gum" whose Quirk is called fat absorbtion. He's a great guy, but it might be offensive

Quirkless people are heavily discriminated against by most of society.

Neither the anime nor the manga have finished airing yet.

Between Todoroki's parents

Todoroki was put through very harsh training as a child, and at one point his mother threw boiling water at him. This has affected him greatly, even as a teenager. Also worth noting: in the manga, a major villain (Overhaul) abuses a 6-year old girl while posing as her "father" figure.

Yes, the character Re-Destro's lungs are injured by Edgeshot, Shemage attacks someone's lungs with fungal spores, and All For One pulls out one of Best Jeanist's lungs.

In the manga a villain is dropped from the sky and falls too their death in a fight

There is a risk of drowning during the USJ arc of season 1, but nobody actually drowns.

In season 3, the villain Mustard uses a gun against Kendou and Tetsutetsu

All Might coughs up blood often due to respiratory issues gained from a fight he was in.

Chapter 160 of the manga