I am flying from Los Angeles to Ho Chi Minh City with my husband and 1 cat, China Eastern Airlines

Your airlines allow pet shipment as cargo. What is the contact tel number to book this cat? What is the rate? The flight is from Los Angeles, CA usa via Shanghai China with final destination to Ho Chi MInh City Vietnam. Is the cat going to be transferred from one plane to the other by the cargo department? Does the cat have to have special transit permit in Shanghai?
Look forward to your reply,
Maria