There is some evidence suggesting that the Romans and Chinese experienced limited contact. The University of Washington provides this document of a Chinese source interacting with the Romans on several occasions. Ptolemy's famous map certainly suggests familiarity with the seas around India. There are various claims of Roman coins found in China, although numismatics are not my thing and I can't verify it. And, of course, every once in a while you'll see something like this discovered.

It's safe to assume that there was contact and trade between (at least) the civilizations on the silk road. Items get around, either through sea routes or by land. The limitations of such routes at various periods is the subject of scholarly debate. As far and foreign policies are concerned, I can only speak from what I know about the Romans, and I don't recall any between themselves and the Chinese.

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Carthage and Rome were likely in contact since the earliest days of their settlements (traditionally dated to 9th-8th century BC)- Carthage was a trading hub for the western Mediterranean and Rome was a significant Italian town connected to the sea by the Tiber River. Carthaginian pottery has been discovered in Rome dating to the city's early period, and the first known treaty Rome made with a foreign power was with Carthage, in the same year as the founding of the Roman Republic (509 BC). Contact kept up as both Rome and Carthage grew in wealth and power over the coming centuries, until they became rivals for control of the region and fought the series of famous Punic Wars starting in the 3rd century BC. Both Rome and Carthage were also in regular contact with Greeks, through the Greek colonies in Italy and Sicily (most of those Greek cities dated back to roughly the same era as the founding of Carthage).

For the Persians, it's a little more complicated. Up to the 6th century BC the Persians were a relatively minor power in southwestern Iran- the rest of Iran was ruled by the Medes, and the Persians were their tributaries. Starting around 550 BC, the Persians rapidly conquered the Median, Lydian, Neo-Babylonian and Egyptian kingdoms, bringing them up to the Mediterranean Sea and to the Greeks and Phoenicians, who they may not have direct contact with prior to their expansion. From that point on the Persian Empire was deeply involved in affairs in mainland Greece and was a key player in Mediterranean trade and diplomacy, although I am not aware of specific instances of their relations with Carthage or Rome.

China seems to have had only a very dim awareness of lands to their west during the Zhou and Qin periods. Geographically, there were two primary routes open to travelers between China and the rest of Asia- over land through the narrow Hexi Corridor and Tarim Basin, or by sea, through the Malacca Strait to India. Both of these routes would be opened up during the Han period, starting during the reign of Emperor Wu around 150 BC. Prior to that, Mediterranean people were aware that there was a distant land to the East that silk came from but lacked any concrete knowledge about Chinese society.

Chinese first learned about the Mediterranean in the 2nd century BC, based on the travels of the emissary Zhang Qian. His travels brought him as far as Iran (then ruled by the Parthian or Arsacid empire) and he gathered reports on Mesopotamia, then ruled by the Greek Seleucid dynasty. At this time the Silk Road expanded greatly and embassies between Chinese, Central Asian, Indian and Iranian states becomes frequent.

We have no indication that Mesoamericans like the Maya had any contact with the Old World at all until the arrival of the Spanish.