Btw as to the TS question, I don't believe there was anything like a trinity in ancient Israelite religion. There was a Canaanite multi-level pantheon of gods, which got systematically collapsed by the Jerusalem priesthood in favor of a single God + his king + his priesthood who deserved complete control and obedience. There were polytheistic elements to this, but they tended to be in the form of divine kingship and divine 'messenger/spirit' of the deity, which sort of oscillates between independent messengers/council and a dependent messenger/spirit. Jesus, as Isa, was clearly envisioned as being in the line of Davidic divine kingship, so there was a polytheistic element to that, but not in the strict trinitarian sense that developed much later.
The Sanhedrin (beit din ha-gadol), at least in their final form, were not identical to the priesthood or the king; the priesthood and king were both, in a sense, under them. They didn't exist yet at the time of the pre-1st exilic kings (although the priesthood seems to have been more firmly established, based on the writings), and there weren't many real kings after that. The Sanhedrin was more or less a supreme court, where they debated the "constitution"--the Torah--and discussed at first what constituted it (though at this time they were mainly still schools of scholars) and later, what each piece meant (at this point they were the hand-picked disciples of scholars), finding evidences from within it to explain how to handle their contemporary problems. In their final form, towards the end of the 2nd temple period, had the power to decide who was a king and who wasn't and what the day to day practices of the Jewish people should be.
Priests had comparatively little power, by that time. In the beginning of the Jewish writings, in books like Judges (which seems to have been written during the time of the pre-1st exilic kings), anyone could be a priest, and it was not limited to a particular clan--although Levites were favored. Judges frequently discusses people setting up their own children as priests. In the kingdom of Israel (once it separated from the kingdom of Judah), there doesn't seem to have been concern over the clan of the priests either: 1 Kings 12:31--"And he made houses of high places, and made priests from among all the people, that were not of the sons of Levi." Being a judge in the pre-2nd Temple era seems to have been something that required consensus of the people, and was a position held for life, but not usually passed on to your children--though it could be passed to your students. There doesn't seem to have been much in the way of earthly qualifications to become a judge before the kings but it seems you had to claim a direct line to God; later, kings could appoint judges, regardless of the judge's link to God, as long as they were pious and had the support of the people.
Judah did apparently have Levite priests, and that became the norm, until the destruction of the 2nd temple, apparently. In the Christian texts it seems to be just a part of accepted practice that the priests were Levites, as the words seem to be synonymous. However, to be a member of the Sanhedrin did not require being a part of a particular clan; you simply could not be a convert, and you must have studied "at the feet" of an existing member, and been appointed his successor by him. A Levite could be a member of the Sanhedrin, but that wasn't automatically true; and a member of the Sanhedrin had to be a Levite to be a priest, being a member of the Sanhedrin did not grant priestly status.