Since the Roman Empire and Republic were so large did they develop any sort of advanced bureaucracy like in China? Were there examinations for civil servants or ministries?
The examination system didn't develop in China until the Tang Dyansty. But yes, the Roman Empire was well known for its massive, highly efficient bureaucracy. In fact the complexity of its bureaucracy is one of the reasons the word "Byzantine" has its definition (describing something incredibly complex, sophisticated, or bizarre) according to John Haldon.
The Late Roman era better represents its sophisticated, centralized bureaucracy than the Republic or the Principate empire would.
Check out this thread here for more: https://www.reddit.com/r/AskHistorians/comments/46w392/the_late_roman_bureaucracy/
EDIT: According to the bot I'm supposed to tag the commentors, /u/hatmaster12 provided quite a sophisticated answer.