One area of Old/New Testament which I’m interested in is the genealogy of Jesus. Now what I’m wanting to find out is whether the genealogy of the line of David (as the Jews know the Messiah must come from his line) is still being recorded to this day? Is there any record of it being recorded from Joseph’s line though say James or one of Joseph’s other sons?. I’ve found a quote of Rabbi shimeon ben Azzai said concerning Jesus “I found a genealogical roll in Jerusalem wherein was recorded such-an-one is a bastard of an adultress”. (Yebamot 4,3;49a). There must have been one around Jerusalem sometime. I thought it might help the case for contemporary documentation for Jesus, atheists seem to want these days.
Genealogy, and especially Jewish genealogies, are not my specialty. However, if I understand correctly, it is true that Jews, and especially those from the Cohen family (those directly descended from Aaron and therefore qualified to be priests in principle) do have genealogies which they purport to go all the way back to the time of Christ and even to times earlier than that. The reliability of such genealogies are up to question, but, in principle the answer to your question is yes.
Of course, because of his death, there is no one directly descended from Jesus. I am not aware if any of his brothers had children or whether there are some who claim to be directly descended from Jesus’ brothers. I am also not aware of whether anyone alive today claims to be directly descended from king David.
By the way, Mormons are particularly interested in genealogies, perhaps even more than most Jews are. They believe in baptisms for the dead, as I assume you know. The “purpose” of such baptism for the dead is so that those who are now alive today as Mormons can prove a direct line from themselves, back all the way to Moses or Abraham, so that they can be accepted into the perfect priesthood. This is the reason that Mormons are so interested in genealogies, as it relates to their false doctrine of baptism for the dead and their concern to have a personal genealogy that takes them all the way back to Abraham and therefor usher them into a higher level of heaven. Sounds crazy, but this is Mormonism!
As for the quote, presumably from the Jewish Mishna, I am not sure if this is a reference to Jesus. However, it is true that Jesus has some “questionable” people in his ancestry such as Rahab who was a person of questionable repute. So, if Jews charge Jesus with having a questionable genealogy–being the son of an adultress, we as Christians accept this charge willingly, as it is in the Hebrew Bible. That he was a “bastard” is a false charge, but Jews see things this way, as you know.
I am not aware of how any of this will affect what atheists and other skeptics believe, unless you can give me a specific example of a criticism of Christianity based on the claims found in Yebamot 4.