In The Curious Incident of the Dog in the Night-Time, who killed Wellington and why? What does the killer's confession reveal about his family situation and the community's perception of the disabled?

In Chapter 167, Christopher learns that it was his father who killed Wellington. His father comes clean, telling Christopher that he will always be truthful with him from now on. He tells his son that after his mother left, he got really close with Mrs. Shears. They formed a relationship and he had hoped that Mrs. Shears might become part of their family. However, their relationship led to lots of fighting. After one really bad fight, Mrs. Shears throws Christopher's dad out of the house. That's when the dog went after him, and Christopher's dad killed it with a garden fork.
This incident shows us that Christopher's father wanted to form another relationship after his wife left. He was hoping to rebuild his family with Mrs. Shears. However, it is implied that they fought about Christopher. Mrs. Shears, it seems, said some very unkind things about the boy. She seemed to care more for her dog than for her neighbors. Christopher's father admits that he acted impulsively out of anger when he killed Wellington.