Does acupuncture work on animals?

Some controlled studies in animals, mostly rodents receiving electroacupuncture, report analgesic effects. That literature is limited by small samples, inconsistent point selection and publication bias, and positive results are more likely to be published. It is suggestive rather than conclusive, and it does not settle the question for humans.

If animals respond, does that prove there is no placebo effect?

No. Animals show conditioning, stress responses and handler effects, and observers who score pain or lameness bring expectations of their own. Removing human belief does not remove context, and context is much of what placebo research actually studies. The inference is weaker than it looks.

Can I use Chinese herbal medicine on my pet?

Only with veterinary advice. Herbs that are safe for humans can be toxic to cats, dogs or horses, and doses are not transferable. Veterinary acupuncture exists as a regulated activity in many countries, and it should be sought through a vet rather than a human clinic.