Short answers, pulled from the story.
Daqin literally means "Great Qin," derived from the founding dynasty of the Chinese Empire established by Qin Shi Huang. The prefix da, meaning "great," signaled that the Roman Empire was considered on par with the power of the Qin dynasty. The name has no phonetic connection to Latin Roma or Greek Romaikē.
The first embassy recorded in Chinese histories arrived by sea in 166 AD, entering through Jiaozhou, the Chinese province now known as northern Vietnam. Its members claimed to represent the Daqin ruler Andun, identified as either Antoninus Pius or his co-emperor Marcus Aurelius Antoninus, and presented gifts to the court of Emperor Huan of Han.
Daqin was the name used from the Han dynasty onwards for the Roman Empire. By the Tang dynasty (618-907 AD), Fulin emerged as a new name, used primarily for the Eastern Roman or Byzantine Empire. The Old Book of Tang and New Book of Tang identified Daqin and Fulin as the same country, though the name Fulin also came to refer to Syriac-speaking Christian communities along the Silk Road.
Gan Ying did not reach the Roman Empire. Sent by General Ban Chao in 97 AD, he traveled as far as the coast of a large sea before turning back after Parthian sailors warned him the crossing was long and dangerous. His written account of Rome was based on second-hand information gathered during his time near the Persian Gulf.
Yes. Warwick Ball, writing in 2016, documented the discovery of sixteen Roman coins at Xi'an, the site of the Han capital Chang'an, minted during the reigns of emperors from Tiberius (14-37 AD) to Aurelian (270-275 AD). A total of forty-eight Byzantine gold solidus coins dating from the reign of Theodosius II onwards have also been found in Xinjiang and across China.
The History of Song described a legal system distinguishing minor from major offenses, with two hundred strikes from a bamboo rod reserved for major crimes. Capital punishment involved placing the condemned in a feather bag and throwing them into the sea, which corresponds to the Romano-Byzantine poena cullei, in which those convicted of parricide were sewn into a sack and cast into a river or sea.