P.M.
Death is such a natural outcome of life that it does our children a disservice to try to shield them from it. Sadness, loss and grieving are part of life, too, and if handled well, they become simply part of a very rich life experience. (And because I have confidence in the rightness of all natural feelings, I think crying in front of children is fine – they just need to know why we are crying, so they don't imagine something far worse than the truth.)
The idea of taking turns can be helpful. When my nearly-4yo grandson's GG died recently, I told him that she loved being with him. But everybody gets a turn at life, and when our turn is up, we die so that new babies (and new animals) will get their turn. For most of us, our turn will be over when we are very old and tired and achy, and have no energy to enjoy our lives any more. Then our bodies quit so we don't have to feel sick and tired any more. This is what's happening to your dog.