In the episode "the idiots lantern" in series 2 there is a a teenage boy who is emotionally and (implied) physically abused by his father, at the end of the episode his father leaves but the boy decides not to cut contact
There are several episodes where one-off characters are shown to be in abusive relationships (which are all portrayed as horrific, and usually end with the abuser facing some kind of comeuppance).
Additionally, there are some scenes where the main characters are shown committing abusive acts towards other main characters they have a close relationship with (including slapping them, yelling at them, and insulting them in a mean-spirited manner). These are almost always played for laughs, and the responsible characters aren't called out for their bad behavior.
S12 E6 Multiple dead birds are shown throughout episode including the dissection of one. Alien bacteria is feeding on the plastic inside the birds. The episode emphasizes the importance of conserving our planet
In the episode Survival, an Earth cat is savaged to death by an alien cat, although this scene is infamous in the fandom for extremely unconvincing special effects, so the blow MAY be softened. Proceed with caution.
A robotic dog that was the pet of Sarah Jane and the Doctor is killed in an explosion. In the original series, the same robot dog accidentally drives into the ocean and short circuits. It is not a particularly lifelike dog but it might upset some viewers.
Arachnophobes will want to avoid the TV stories Planet of the Spiders, "The Runaway Bride", and "Kill the Moon", as well as the novels "Festival of Death" and "The Crawling Terror" and the audio drama "Empire of the Racnoss".
No but in the 10th Doctor episode "The Girl In The Fireplace" and then multiple times in the 11th Doctor's series, The Doctor ends up being kissed by or is in a romantic relationship with women he first met when they were children. It's an odd reoccurring theme in Moffatt's writing.
Planet of Fire involves a character burning to death on-screen. Dark Water claims that humans are conscious postdeath and spend their final moments mentally begging not to be cremated; this turns out to be false.
Series 5 Episodes 8/9 (The Hungry Earth/Cold Blood) shows multiple characters being sucked underground by the earth, which leads you to believe they’ve been suffocated. It is later revealed that they weren’t actually suffocated in the earth, and they’ve been taken captive by an underground civilization. Nobody is actually buried alive, but at first it may seem like they have been.
**Spoiler warning for planet of the Ood** we watch a human become an Ood and its nasty. To skip, about 40 minutes in when The Doctor says "Oh they've been preparing you for a long time" to Halpen skip ahead by about one minute.
The Tenth Doctor has his hand cut off and the hand in a jar appears multiple times. The Twelfth Doctor injures both hands punching a wall in the episode Heaven Sent. It is shown MANY times in the episode.
Many aliens, both minor and major characters, die. The Doctor is an alien who looks human, as are other Time Lords and Ladies, and they die in some season finales, too.
If regeneration counts as "death" the Doctor and Master have died A LOT. Even not counting regeneration, there are moments (and alternate time lines) when both of them are stated to be dead and there are some wild plots to resurrect them. Several companions have either died or come so close it basically counts.
While Trypophobia is commonly known to be the fear of holes, it can also be triggered by similar patterns and clusters of bumps. In "The Woman Who Fell to Earth" there is a character with teeth implanted in his face which may trigger this.
aliens in london/world war three/boom town feature the slitheen, who are aliens inside of human skin suits and often “have issues with the gas exchange” (leading to the iconic line “do you mind not farting while im saving the world?”)
in "The Long Game" there's a scene where Adam, after receiving his upgrade, vomits from the shock of seeing his own brain. However the vomit itself is turned into an ice cube.
In S3 E2, the Shakespeare Code, the Doctor, Martha and William Shakespeare go to an asylum to talk to a patient there and the people there are treated incredibly badly.
Episodes involving hospitals include The Empty Child/The Doctor Dances, New Earth, Smith and Jones, The Eleventh Hour, and World Enough and Time/The Doctor Falls. I think that's all of them, but I could be wrong. The Curse of the Black Spot and The Girl Who Waited both involve hospital-like locations, but they are not actually hospitals.
Not explicitly but the doctor is sometimes accused of lacking empathy/emotion for not expressing it normally. Characters are teased/othered for being “weird”/“different”.
Amy, Rory and The Doctor are under the influence of "dream lord", a being who traps people in false realities. The three of them spends the whole episode trying to figure out what place is real
The Master is haunted by the 'drums', a repetitive series of taps, which the audience hears in the season 3 finale, The Sound of Drums/Last of the Time Lords
In "Water on Mars" [SPOILERS] the Tenth Doctor attempts to save a group of scientists on Mars but upon returning them to Earth, the leader immediately kills herself upon walking into her house.
Not technically an eating disorder, but Season 4 Episode 1 features a weight loss drug, and several times throughout the episode characters point out how much thinner/better they feel taking the drug and loosing weight, which may be upsetting to some viewers dealing with eds.
not in a suicidal manner, but characters do intentionally harm themselves on several occasions, such as one character breaking her own wrist to escape a trap.
There is a scene shot with the camera close to the water level, and a wave travels toward the camera, which is briefly underwater before the shot ends.
aforementioned voyage of the damned episode: there's LOADS. there's two 'comically fat' characters and a lot of the stuff surrounding their characters concerns the fact that they're big. there's also a character actively fatshaming them constantly
there are no slurs used, but there is some mild homophobic language from the master directed at captain jack harkness (although, time lords dont share human ideas of gender and sexuality, so take that as you will)
An Alien character (Strax) who doesn't understand human gender cues often/constantly uses He and She for the wrong characters (Humans who are presumably cis). This is presented as funny.
There are relationships between humans and aliens and different species, like Jenny and Vastra or Rose and the Doctor. However all parties are intelligent species and can give full consent.
There is a scene with Rose while she is hanging from the air that someone makes some uncomfortable comments about her, once she is no longer hanging it stops.
Davros is dying, and indeed very close to death, in the episodes "The Magician's Apprentice" and "The Witch's Familiar." It is never stated what exactly is killing him, but he is very weak and looks and sounds ill.
In the episode "The End of Time" a few unlucky homeless individuals are shown receiving food from a charity. They are murdered soon after by the Master.
In the episode "The Doctor Falls," the Twelfth Doctor gives a magnificent speech about finding purpose in life and in death. "Hey, you know, maybe there's no point in any of this at all, but it's the best I can do, so I'm going to do it. And I will stand here doing it till it kills me. You're going to die too, some day. How will that be? Have you thought about it? What would you die for? Who I am is where I stand. Where I stand, is where I fall."
By far most episodes have a happy ending, either on TV or any other medium, with the Doctor saving the day with minimal loss of life, before, after being thanked by the survivors, going off to another adventure.
However, there are some rare exceptions to this rule, including, but not limited to, The War Games (the Doctor is successfully captured and executed by the Time Lords, before regenerating), Doctor Who and the Silurians", The Sea Devils", "Warriors of the Deep" (in all 3 cases, the Doctor fails to make peace between humans and an intelligent reptile race called the Silurians, resulting in a huge death toll for both species) "Blood Heat" (novel, the Doctor is forced to destroy an entire alternate universe), "He Jests at Scars" (audio drama, the Valeyard and Mel get trapped in suspended animation for all eternity, forever concious but unable to move or even scream), "Lucie Miller"/"To the Death" (two-part audio drama, three of the Doctor's companions die, driving him almost to insanity), and "Army of Ghosts"/"Doomsday" (two-parter, the Doctor is forcibly seperate from his companion Rose Tyler).
As I said, this is not a complete list, as there are probably more I've forgotten, not seen/read/listen to, or you may disagree on whether it counts as 'happy', but MOST stories not on this list SHOULD be safe.
Guns are scene/used multiple times throughout the series, but it should be noted The Doctor has a deep hatred of them, refuses to use them, and distances himself from those who do.