No, BUT a wolf is shot at around 13 minutes or so. But it's not killed. It's injured. It looks like someone is going to beat it later but they part ways.
PTSD is evident in a lot of the characters in mostly small ways. One bigger one, both trigger warning and spoiler warning:
Years after being raped, Cornelia has a consensual encounter and begins to kiss someone she has feelings for. She gets triggered and runs away, takes off most of her clothing, and scrubs her skin with sand for a long time while sobbing. It was such an accurate portrayal of what a sexual assault trigger can look like, i found this scene more triggering than the scene after she was raped (offscreen, but she is shown right after.) But that might be because i knew that scene was coming and this one caught me off-guard.
A character takes off her goggles at the beginning of the 5th episode to show the protagonist that her eyelids have been amputated.
In the 6th episode another woman suffering from stage four syphilis reveals her face that has been severely disfigured by the illness, and there’s another character who seems to have lost an eye to syphilis, too.
Not black, but a minority. The series opens with a bloody body on the ground, clearly dead, though the face is unseen. A native woman is grieving and striving to get to him, and it is made clear that it is a native man who has been killed. The soldiers shoot the body several times, unnecessarily, and that goes on through the whole scene.
Episode 1. A horse gets shot and it is implied that Emily Blunts’s character shoots a pig at the end of an episode to prove she can shoot. You don’t see the pig get shot or see the dead animal
A Cheyenne war chief is shot in front of his wife and son in episode one.
In episode two a man is killed on screen running towards his pregnant wife who’s resting in their wagon. We later learn the wife was killed, too, but the unborn child can be saved and there was a girl hiding inside the wagon who also will be delivered to their relatives later on.
Most characters on the show are racist. Expressions of that racism range from period-typical white supremacist attitudes to committing acts of genocide. None of these attitudes or acts are shown from an apologist angle though. On the contrary, most of the villains are almost cartoonishly evil.