Joseph, there is no evidence or any reason to believe that the Ethiopians known as “Beta Israel” have any Jewish ancestry at all. Those who have become gerim in the last 40 years are of course Jews; the rest are not.
It is completely obvious and not even slightly controversial that if someone discovers that his maternal line does not trace back to a Jewish woman then he is not Jewish, and if he wishes to be he must become a ger. However, what makes you think there are a significant number of such people? How exactly would “female non-Jews erroneously got mistaken for Jews”?
It is also completely obvious and uncontroversial that there are many Jews in the world who have no idea that they are Jews, because their Jewish ancestor in the maternal line married out many generations ago, and that these people are 100% Jews, just like any of us. There were many many gerim in Roman times, most of whom eventually became Xians. Their female-line descendants are Jews. More recently, until 500 years ago the south of Italy was full of Jews; then Spain conquered it and brought in the inquisition, and suddenly there were no Jews. Many of the Jews fled, but most probably remained and became Xians. Also in Spain itself, the majority of Jews remained and became Xians. All of their female-line descendants are Jews. There is nobody who disputes this.
When Eliyohu informs them of their yichus they will have to start keeping mitzvos, and will have to bring chato’os for the sins they did beshogeg. The gemoro says that a Jewish child who grew up not knowing that he is obligated to keep Shabbos must bring one chatos for all the Shabbosos that he violated, because they were all included in the same “he’elem”.
we know that Rashi and the Baalei Tosfos shtam ben acher ben from Dovid HaMeclech, hence Shevet Yehuda.
No, we don’t know that at all. We don’t even know that Rashi was descended from him at all (though it seems very likely), but we certainly don’t know that he was ben achar ben. (And of course nobody is ben achar ben from Rashi.)
We have the complete yichus brief from Dovid HaMelech through Rashi.
No, we don’t. No such document exists, and there is no reason to suppose one ever existed.
YRST, I doubt your friend can actually trace himself to Shevet Shim’on. More likely he simply has a family tradition that they are from that shevet. Is that tradition authentic? Who can possibly know? It is certainly possible. And since one of the major opinions in the gemara is that the ten tribes’ exile only lasted a short time, after which they returned to Eretz Yisrael and then shortly thereafter went into Galus Bavel, there’s no reason to be surprised if a group of Jews from Shevet Shim’on settled in a certain place and their descendants still have a tradition about it. Of course this is not proof, and even it it’s true, over the course of 2500 years many other Jews would have moved to the same place and assimilated into their community, so someone whose family is from there now is not necessarily descended from the original settlers.