SHANGHAI, May 17 — Weeks after tainted Chinese pet food ingredients killed and sickened thousands of dogs and cats in the United States, this country is facing growing international pressure to prove that its food exports are safe to eat. But simmering beneath the surface is a thornier problem that worries Chinese officials: how to assure the world that this is not a nation of counterfeits and that “Made in China” means well made. Already, the contamination has produced one of the largest pet food recalls in American history, heightening global fears about the quality and safety of China’s agricultural...