It is possible to send animals overseas. You would need to look at the current airline policies and there will be paperwork and a quarantine period and extra costs charged by the airports/airlines, but it is possible.
If you really need your pet with you in China, then it is best that you wait until you are settled and then have them come over.
Getting a pet in China is also very easy and many of the teachers have cats or dogs.
Hotels/Apartments allowing Pets
For pets, asking for permission when they cannot see the animal may be an issue. Also arriving and staying in a hotel while you search for your apartment, also will be an issue for most hotels.
We haven’t had any issues with landlords objecting to teachers getting a new pet, but of course it is best to ask about the policies before getting a pet there.