Short answers, pulled from the story.
Daqin literally means Great Qin and refers to the Roman Empire or specifically Syria as known by Han period Chinese. The prefix da signified that the Roman Empire stood on par with the might of the Qin dynasty.
The first embassy arrived by sea in 166 AD via Jiaozhou claiming to represent the Daqin ruler named Andun. This name likely referred to either Antoninus Pius or Marcus Aurelius Antoninus.
Warwick Ball noted the discovery of sixteen Roman coins at Xi'an minted during the reign of various emperors from Tiberius to Aurelian. Only forty-eight gold solidus coins from the Eastern Roman Empire have been found compared to thirteen hundred silver coins.
An embassy from Fulin arrived at the court of Emperor Taizong of Tang in 643. This group claimed to represent their king Bo duoli possibly Constans II.
Chinese histories describe a form of capital punishment involving stuffing the guilty person into a feather bag and throwing them into the sea. This corresponds with the Romano-Byzantine punishment of poena cullei where those who committed parricide were sewn into a sack.