Yes, I am pretty sure it is the same child. Most hospitals encourage families to spend time with an infant that is either stillborn or dies shortly after birth. While it is unusual they took the dead child home vs bringing the others to the hospital, it isn't that big of a surprise.
I would spend time with the child, I seriously doubt I would expose my young children. There was a time in our society that death was dealt with at home, but times have changed. It is a personal matter, but I would not do what they did. I two families I knew who had third trimester children that died. They took pictures, dressed the baby, said their good byes, but they did not take the child home to the other younger child. Maybe, we should, as a society be more at ease with death, but we aren't. I had nightmares for several months after my grandmother died. I was 10 and was permitted to go to the funeral, I decided after that not to attend anymore funerals until I was in my 20's.
To each his own, but I would not be comfortable exposing my living children to my dead one, but one does not know what one would do until it happens to you. I just find it a bit much.