Yes, China has courts and there are approximately 3,581 courts in mainland China.
China Judiciary System 中国司法体制
Zhou Qiang (周强) is the current Chief Justice of China, that is, the President of the Supreme People's Court.
The National People's Congress (NPC) elects and dismiss the President of the Supreme People's Court (SPC).
Yes, China has a Supreme Court. The full name of China's Supreme Court is the Supreme People's Court (SPC, 最高人民法院).
China's judiciary comprises of courts, procuratorates, supervisory commissions, and two other governmental organs—public security organs and judicial administrative organs.
The PRC Organic Law of the People's Courts(人民法院组织法) helps us to gain a better understanding of China's court system and the internal structure of Chinese courts.