You're posting right before Shabbos (the Sabbath) so you're not going to get too much response till tomorrow night, but since I'm avoiding cleaning desperately by being on MDC, will briefly briefly respond ...
First of all, the "European conversion" ca. 900 CE refers to the conversion to Judaism of the Khazar king and his people. A very famous Jewish book, known as the Kuzari, was written as the a Jewish rabbi's explanation of Judaism in response to the Khazar king's questioning during the king's spiritual search.
Anyway, there is a fascinating canard popular among various anti-Zionist groups that all Jews today are all descended from this Khazar kingdom.
First, where did the king get his answers from (and where did we get that book from) if there were no other Jews in the area?
Okay, that was being wise-acre. But it's really one of the more amazing conspiracy theories that have been woven around Jews ... that they're not really Jews at all.
Hebrew Israelite groups are a complicated question, but suffice to say many of them are Christian groups that have found their way back to Ethiopean origins, etc., and are halakhically (according to Jewish law) not technically Jewish. Meaning, their mothers are not immediately descended from a Jewish woman (Jewishness goes through the mother).
Technically to be accepted Jewishly (even among Sephardim) they would have to convert.
The Ethiopean community has had great difficulty with this issue in Israel, as they've been made to go through conversions, despite many being able to point to matrilineally Jewish lines. That's an entirely different issue than the Hebrew Israelites, who may claim that generations ago their families were Jewish ... but the generational remove is also a Jewishness remove, IYKWIM. And some of the groups are actually Christian Messianic groups that feel connected to Judaism just because Jesus was a Jew. Well, that doesn't make anyone Jewish. That makes them Christian.
Each group has a different background, so ...
Okay. That was totally unclear, totally rambly, and totally chaotic for an answer, probably makes no sense, but my khallah is done rising and has to be shaped and put in the oven so more after Shabbos ...