1. Toi, we are dealing with people who either have undergone giur or are children of those who have (the main Beta Israel aliya occurred almost forty years ago although there were aliyot before and after). The Eida does not recognize anyone outside their own little circle as being real observant Jews.
2. I saw three hechsherim on Barkan wine: Rabbanut, Badatz Bet Yosef (established by Rav Ovadia – the Bet Yosef is machmir on the percentage of water allowed and once at a wedding RO said “shehakol” on the wine) and Chug Chatam Sofer (Chareidim, mainly Chassidim and mainly in Petach Tikva and Bnei Brak). As Barkan is sold internationally I presume that the other two are non-Israeli and thus do not appear on Israeli labels.
3. I have already listed the kullot that come from this chumra. Scroll up.