There seems to be an overall confusion in this post over the modern House of David.
To begin:
1. Davidic lineage is very well-documented amongst Jews, we know who is of the line of David and who is not. For most of history following the destruction of the Temple, the Jewish people were led by Exilarchs (the Reish Galuta), these individuals were known as Nasi (Prince), they held a political position, and at various points in history, even had autonomy for the Jews of Babylonia.
2. Most descendants of David are actually Ashkenazi, because Rashi was of the line of David, and many Ashkenazi people trace their line back to him. One well known Davidite of modern times was the Lubavitcher Rebbe of blessed memory.
3. There are no formal "families" per say of Davidites, and unlike the Priests/Kohanim, they usually didn't adopt special surnames to show their status. The number of Davidites is usually given around ~600,000, the number of Priests ~500,000, and then number of Levites ~2,000,000-4,500,000. (the first number representing confirmed Levites, the latter being patrilineal Levites who either don't know they are, or only have a Levite surname [Levin/Levitz/Levi/etc].
The rest of Jews belong to the other Tribes, I for one, am of the tribe of Benjamin.
In some rare cases, where male converts were accepted (which was rare, due to persecution), the child and all his descendants would thus not be Israelites, and wouldn't have a tribe. It is impossible to know how many of the non-identified tribe Jews are truly patrilineally converts, simply because all Jews, irrespective of descent are considered descendants of Abraham.
We also have evidence many Jews matrilineally descend from the tribe of Shimon/Simeon, this of course, doesn't mean they belong to that tribe. Just that, due in large part to the near extinction of it, very few people can actually claim descent from Simeon. There's a small possibility, that during the Roman period, the remnant of Simeon were considered part of the tribe of Judah, meaning even male Judahites, could be theoretically of the line of Simeon, but of course, this would be only in the cases (rare as they are) where Jews don't have their full genealogical records.
I for one, not only know the names of all my ancestors, but also their locations and in some cases (especially the last 400 years) their secular occupation [if they had one].
I come from a Rabbinical family, thus I have many Rabbis in my ancestry.
As to who Yehudah HaNasi is? He is the compiler of the Mishnah, and lived around 135CE, the same time as the Bar Kochba revolt against Rome. His name in English is "Judah the Prince", and he is regarded as a great sage of the Mishnaic period.
