I didn't knew Zacharias was a Sadducee or that only priests were Sadducees.
Sadducees claimed to be the descendants of Tsaddoc, the High Priest from the time of David. They were the conservative aristocratic party to whom most
of the clergy of the Temple belonged. They believed in the Torah alone and refused later doctrinal developments such as belief in the next world, resurrection of the dead, heaven and the gehenna, angels etc. This made them oppose the Pharisees. Juridical power on religious issues was entrusted to both parties in the Sanhedrin Court.
Zacharias, the father of St. John the Baptist, wasn't necessarily a Sadducee because he was a priest.
So He would have been a bit of all?
No - he might have incidentally agreed or disagreed with some. For instance he contradicts the Sadducees, agreeing with the Pharisees, that the God of Abraham, Isaac and Jacob is the God of the living, not of the dead, that there is resurrection, judgement and retribution, the Kingdom and the gehenna.
Did He have any connection to the house of Hillel(Beit Hillel) , the house of Shammai(Beit Shammai) , or other Jewish trains of thought?
He would have been closer to Hillel (they were more Gentile-friendly, more lax in application of the Law) on some issues, and closer to Shammai on others (divorce, IIRC). There's no proof that he would have had personal contact with either school.
What were the circulating trains of thought among the sects of Judaism in the 1st century and what soteriology did they have?
Couldn't possibly answer that one in a post - probably each believed you had to be one of them to be "on the right path". The Sadducees didn't believe in an after-life, so salvation wouldn't have been much of an issue for them. God would bless you in this life, if you obeyed the Torah, would not punish you for sins if you atoned for them through the prescribed sacrifices - once you died, you disappeared.
Salvation back then would have been more of a collective issue than an individual one: the earthly restoration of the Kingdom of David under the Messiah. The Pirke Avot (a section of the Talmud) begins with the bold assertion that "All Israel will be saved".
What would have been the trend among Jewish believes?
Unfortunately, we do not have any statistics to establish the following of each sect.
When were these Jewish sects born/separated in the form of Pharisees, Sadducees, Essenes,Zealots, Samaritans,etc and what common ground did this sects have on the time of Jesus?
The split occurred when the land was conquered by the Greeks (312 BC); the Pharisees were the descendants of the "pious" (chassidim
) who zealously upheld Jewish traditions and resisted Hellenistic assimilation. The Sadducees and the priestly aristocracy reached some settlement with the foreign rulers, so as to allow Temple worship to go on. The Essenes are supposed to be the followers of a dissident priest (the Teacher of Righteousness) who retired to the desert to establish a spiritual Temple and considered the guys in Jerusalem to be apostates and impostors. The Samaritans were the result of the Assyrian conquest of the Northern Kingdom (722 BC) - that would have been the earliest schism in chronological order. Zealots were revolutionary guerillas that appeared under Roman rule.
Common ground? Maybe the Torah, which each sect interpreted differently. The Temple in Jerusalem was frequented by most Jews, with the exception of the Essenes and the Samaritans.
But all this you could have googled up or read about on Wikipedia.