The dog is shot off screen. However leading up to that, the dog is in the vehicle when they are in a collision and is injured. You hear the dog crying and whimpering for an extended time prior to it being "put out of its misery".
No, but it does get hit by a car. We see the main characters riding in the car, the woman distracts the husband, who is driving, so he crashes onto a horse. The scene is loud and dramatic. The horse runs away quickly, but we see the front of the car is stained with blood. Later in the movie we see the horse again and it has open wounds in its face.
I would say that there is a particularly notable scene where LJA is at MIL's house where there is a lot of flies. You can see them occasionally, but the volume is very loud and she is trying to kill them with a fly swatter.
There's a scene where a character takes a baby into the woods for a few hours without telling any members of the family where they are, which causes distress, but the baby is brought back safely soon afterwards
The wife throws herself through a glass door and gets cuts all over her, she hits her head on a mirror at one point, and injures herself while destroying her bathroom
I would say arguable, and that there is ableism perpetuated by the directors and writers, but not specifically from the characters.
It is shown that a man has a disabled daughter and that the daughter has become a severe strain on his life/marriage. Just disappointing to see this idea in a movie time and time again, would love to see just a normal family with a disability included, not having the disabled character be a burden to able-bodied people.