Aramaic was the official language of the western portions of the Persian Empire.
As to Idumaea, after the Babylonian conquest, the Edomites had settled in the southern part of Judah, especially west of Hebron. They were fleeing from the Arab conquest of their original homeland, Edom (south and southeast of the Dead Sea). Their new settlement area was later called Idumaea in Hellenistic times but was distinct from the original Edom. The Edomites' original language was a Canaanite tongue close to Hebrew. They were converted to Judaism in Hasmonean times and were considered Jews from then on. King Herod was a Jew of Idumaean descent.
Palestine obviously derives from Philistia (Pleshet in Hebrew), the southern coastal area of the country. The great French scholar, Victor Bérard, fighting to stem the aryanist tide in classical studies earlier in this century, saw the names Palestine and Syria as both resulting from the same phenomenon. The names originated among merchants and sailors coming from the West. They came first to coastal cities. Abel has described the phenomenon as follows:
Thus from the territory of the Philistines, the name Palestine was extended to the whole hinterland of Southern Syria. By a process familiar to the ancients, the name of the closest, most accessible tribe was applied to the whole country.