Edomite King Herod Agrippa

Is "Herod" (Acts 12:18) the Edomite king who tried to kill Jesus after His birth?
No, this is King Herod Agrippa I ("Agrippa"). The King Herod who killed all boys two years and under in Bethlehem of Judea after the Birth of Jesus (see Slaughter of the Innocents) was called "Herod the Great" for the great buildings he built, including the second temple of Jerusalem. After he died, his territory was split into three, each to be governed by his sons, Archelaus, Antipas and Philip (see Archelaus). Agrippa is the grandson of Herod the Great and the son of Aristobulus IV, who was killed by his own father (Herod the Great). Agrippa grew up in Rome and endeared himself to successive Roman emperors. Emperor Caligula made Agrippa king over the territory of his uncle Philip in 37 AD, then gave him the territory of his uncle Antipas in 39 AD. Emperor Claudius gave him Judaea and Samaria in 41 AD, at which point Agrippa became the ruler, albeit still under the thumb of Rome, of more or less the territory of his grandfather. Agrippa died in 44 AD (see below), to be succeeded by his son Agrippa II (see King Agrippa), so the events in this chapter took place between 41 AD and 44 AD.

Why did Agrippa have the guards "put to death" (Acts 12:19)?
If the guards were truthful, they would have told Agrippa that they have no idea how Peter disappeared, which would have been incredible to Agrippa, who would have surmised that the guards either had colluded with Peter to let him escape or had fallen asleep while on duty. In addition, Agrippa most likely was angry and wished to take it out on someone. After all, he had awoken that morning with the intent to kill.

Why did Agrippa then go "down from Judea to Caesarea" (Acts 12:19) and stay there?
Had he killed Peter, Agrippa would have demonstrated his power and won accolades from the Jewish leaders, who would have feted him for helping them crush the local church. By arresting and then losing Peter, Agrippa had instead demonstrated his impotence against and emboldened the local church, much to the displeasure of the Jewish leaders, who may have given him the cold shoulder for making matters worse for them. What better place for Agrippa to go and lick his wounds than Caesarea, where the governor and the legion of his Roman overlords were based?

Who ruled over whom?
The Jewish ruling council - Sanhedrin - was under the authority of Agrippa and the other Herods, who were Edomites. The Herods in turn were were under the authority of the Roman emperor (see the explanation under When Jesus was Born and Augustus Caesar).

Who are "Edomites"?
Edomites were the descendents of Esau, the son of Isaac and the older twin brother of Jacob (see Patriarchs): "Now this is the genealogy of Esau, who is Edom. Esau took his wives from the daughters of Canaan: Adah the daughter of Elon the Hittite; Aholibamah the daughter of Anah, the daughter of Zibeon the Hivite; and Basemath, Ishmael’s daughter, sister of Nebajoth. Now Adah bore Eliphaz to Esau, and Basemath bore Reuel. And Aholibamah bore Jeush, Jaalam, and Korah. These were the sons of Esau who were born to him in the land of Canaan. Then Esau took his wives, his sons, his daughters, and all the persons of his household, his cattle and all his animals, and all his goods which he had gained in the land of Canaan, and went to a country away from the presence of his brother Jacob. For their possessions were too great for them to dwell together, and the land where they were strangers could not support them because of their livestock. So Esau dwelt in Mount Seir. Esau is Edom. And this is the genealogy of Esau the father of the Edomites in Mount Seir" (Genesis 36:1-9). "Mount Seir" is the mountainous region south of the Dead Sea.

How did the Edomites come to rule over the Jews?
See King Herod and Archelaus.