Most characters struggle with their mental health. One of the main characters is believed to be losing his mind by everyone around him, although he's not.
I don't recall anyone being hung by the neck (though I can't 100% swear it doesn't happen), but there are vivid descriptions including a whole chapter involving one of the main characters in which people are hanged from a building upside-down by their feet and left to die.
One of the main characters contemplates suicide repeatedly, and attempts it once. There is also a location where it is stated that numerous unnamed characters have committed suicide.
There is a supporting character that is strongly implied to be on the spectrum who is treated poorly by his community and bullied. He is eventually killed, though this is in battle and has nothing directly to do with him being on the spectrum.
Yes, but not as a direct result of their mental illness. Kaladin is a soldier with depression and PTSD. Shallan also has a mental illness that is further explored in later books.
A supporting character with PTSD (and possibly a head/brain injury) is treated badly by the people in authority, and initially by some of the supporting protagonists, though one of the main characters treats him well and brings the other protagonists around. There is another supporting character who is missing an arm who is initially treated with disdain.
There is a relationship between a man and his brother's widow, which is not incestuous by the standards of our culture but is described as incestuous by the standards of their fictional religion.