The newspaper notice on Cora's death specifies that the driver wasn't blamed for the incident. Why? Because it's implied she ran out of the house into the coal cart's path because Joseph had frightened her somehow.
Why does the normally stoic John faint after learning how Joseph died? It's because Joseph's situation is the exact antithesis of the situation that has been tormenting John. John has been agonising over what he could have done differently to save his daughter, whereas Joseph's father actually murdered his son himself. It's darkly ironic and horrifyingly incomprehensible to John how someone could do the very thing that John regrets not having been able to prevent. No wonder the revelation hits John particularly hard - he looks nauseous before he faints.
This also explains why Joseph's spirit latched onto John. The Medium senses that it's John's recent tragic loss that attracted Joseph to him, but it's been over fifty years since his death, surely he encountered someone else dealing with grief in that time. But John is specifically grieving over his failure to save a child, and Joseph is a child in need of saving.