Shipping your pet from US to Dubai...
You're better off using one of the agents mentioned above vs doing it yourself. They specialize in it and know all the pitfalls.
Use a Dubai based agent, and they will forward you their preferred agent on your side. It will cost extra but will save you heaps of running around. They would also have a preferred airline that has a good track record of shipping pets.
We followed this route and weren't disappointed.