There is at least one instance of DV shown by the victim's wounds and a description of her experience in a criminal trial in s2e2. Numerous main/secondary characters discuss their misogynistic opinions of the period typical DV in that same episode.
S2e20 has the new teacher come to town. She uses physical abuse to try to get students to behave and focus. You don't see most of it happen, but you see the aftermath.
In s2e16 Myra, a sex worker tells Hank that she doesn't want to have sex, and he doesn't take no for an answer. It is assumed that after the screen fades to black that they do indeed have sex.
In s2e16 Myra, a sex worker tells Hank that she doesn't want to have sex, and he doesn't take no for an answer. It is assumed that after the screen fades to black that they do indeed have sex, but it is not on screen.
The head of the local kkk sets himself on fire accidentally in s2e20. He's fine after (well he needs to get his arm looked at, and Dr. Quinn offers because she's a good person, but he turns her down).
In s1e4 there is a teenage boy who a mob is set to hang without a trial. The noose goes around his neck before someone else cuts the rope. Before that scene, there is a whole scene where members of the town are self righteous about putting up the rope and preparing to hang the boy. In s2e20 the kkk tries to lynch Robert E, and the rope is cut by Sully. (honestly, I recommend not watching this episode because it feels a bit like trauma p*rn. There is a great conversation between Robert E and Brian, but it's incredibly upsetting to watch the rest of the episode.)
In s2e16 a 15 year old goes to Hank at the saloon to work for him as a sex worker. He tells Myra to get the young girl ready to work. Myra ended up getting her out of her contract, so she never ends up having to work at the saloon.
While there are no scenes in a modern hospital that might trigger a person (seeing as this takes place in the late 1800s) it does follow the lead as she practices medicine in the frontier. And that is explicitly shown.
In s3e10 Dr. Quinn finds a lump in Dorothy's breast. Viewers find out Dorothy's mother died of breast cancer, and Dr. Quinn urges Dorothy to have the lump removed. It results in her subsequently having one of her breasts removed. In s3e16 a character comes to town because she had leukemia and plans to die in the mountains.
If you count addiction as said mental illness, yes. In s2e19, Dorothy's son who is addicted to morphine is violent in his attempt to get more from Dr. Quinn.
It is not explicitly stated as that in the moment, but a character tries to drink himself to death and then talks about it later in the same episode (s1e12). There are other instances of attempted suicide as well (s5e21)
So not exactly, but it can be similarly triggering. In s1e12 a character feels so guilty about something that he drinks so heavily he almost dies of alcohol poisoning. After he recovers, he expresses that he had wanted to die and that had been the way he'd tried.
Off camera a character who exists for only two episodes in season 3 and is an antagonist during them is mentioned to have killed himself in the last few minutes of s3e8 very briefly. It's kind of sudden and then it's brushed past. And in s5e21 Horace attempts suicide, but he survives and decides to go away for some time.
Throughout there are characters who have children and then carry them around in scenes as they get older (and sometimes are sick and need medical attention)
S3e13 is a holiday episode in which Jewish immigrants come to town and make the Christian community question their holiday spirit. I cannot speak to the specifics because it's an episode that I skipped out of avoiding the potential of antisemitism.
In s3e11/12 Jake and Loren wear feminine clothes to drink free for ladies night at the saloon. Jake is mistaken for a woman when the saloon is robbed and taken as a hostage. It is played off as a joke that the kidnappers think they have a woman for the two episodes.
In s5e8 Jake's father comes to town for the first time since the show began and doesn't remember his son. Dr. Quinn sees signs of dementia in him. He is only in that one episode.