Ujm,
1) They are in the same location in Ethiopia, they match the description of those sources, and we can track their basic history from that time. There are also no other candidates. There’s no other group in that part of Africa even claiming to be Jewish. If you say the Beta Israel are some new group, then you need to explain what happened to the Jews of Ethiopia and where their replacements came from.
2) The Islamic sources that I have seen cited are from the 16th and 17th centuries. The Ethiopian Jews are called Falashas (which the Beta Israel are still called by the Ethiopian gentiles) in these sources. There are Christian sources going back to the 4th Century. Around the 12 century or so, after the rise of the Ethiopinum empire, the Christian sources also begin to refer to the Beta Israel as Falasha.
3) I found it online here at http://www.ybz.org.il/_Uploads/dbsAttachedFiles/Article_110.8.pdf It was published in an Israeli academic journal called Te’uda. The letter is from 1862. But references earlier communications. As for how I know it is the same people, there’s no other candidates. How do you know that current Chasidim are descendants of the Talmidei Baal Shem Tov and their followers? Maybe all the chasidim got wiped out in 1870 and new people arose (out of nowhere) and started pretending that they were always chasidim.