One of the prominent theories regarding this is that fairies die after their purpose is fulfilled. Hence why they disappear after healing. Their purpose was healing, they heal you and then die. This is usually used as a theory for what happened to Navi after OoT. I'm not sure I buy this, but its an idea. As far as them being part of a Kokiri, I don't believe they are one in a soul-sharing kind of way, but more that the Kokiri and their fairies spend so much time together they become very attached to each other. That way they are able to help and be an extension of each other's will, because they know each other that well.