Are we re-united with our pets in the Spririt-world?

Yes, because they have an individualized existence in spiritform and because they are drawn to us in the afterlife through the Law of Affinity. Wild creatures not having an intimate association (that is to say an association where emotions are involved) with a human being, will reincarnate almost immediately, but pets will normally remain in spiritform until there is a reunion. Occasionally a pet will reincarnate relatively soon and come within the orbit of those with whom it has an affinity on Earth. The principles under which this operates are complex but known.