Yes, Jesus' reference to the ox in the ditch was a response to a challenge regarding the issue of healing on the Sabbath.
It's 1:30 in the morning here, so a more complete response will have to wait til tomorrow, I'm afraid.
Regarding the Essenes it is extremely unlikely that the majority of early Xians would have come from their ranks, as I understand it.
The Essenes were a hermetic end-times cult. They lived apart, outside of Jerusalem, and had their own cult scriptures in addition to Biblical scrolls.
Xianity, by all accounts, was primarily a religion of urban Jews and of converted gentiles.
I've seen some attempts to equate early Xians with Essenes on the basis of certain similarities in doctrine and terminology, but these are much more easily explained by the fact that both were apocalyptic Judaic cults in the same time and place. Of course they shared many common terms and practices.
It's also been argued that Jesus must have been an Essene because he wasn't a Pharisee or Saducee, but this ignores the fact that the Jesus cult was its own school of thought.
When you read Essene documents -- which is possible to do now -- you do find a lot of the same language, but this was language that was all over the place back then, and there is absolutely no indication that the "son of God" or "son of light" of the Essenes was Jesus.
Instead, these references -- and others outside the Essene scriptures -- inform us that the terms applied to Jesus were common in their day.
And I don't know of any evidence that early Xian writers and preachers were targeting Essenes. You could hardly classify those guys as "low hanging fruit". The Essenes were already walking the walk, big time. They were out there living in the desert, waiting for the end of the world.
Their silence about the Essenes is more likely due to their having no contact with such a remote and insular cult which, while also apocalyptic, already had its own annointed one and its own doctrine.