Wow. So much to say here.
DY: As far as I know, R’ Soloveitchik gave no such Heter. R’ Schachter says he used to refer to a Rabbi who took a job in a Conservative Shul as a Mumar L’teiavon. Also, back in the day there would have been an actual Kaballas Ol Mitzvos by some. Not all, not many, but some.
Lior: The basic tenets of most of the Conservative movement were K’firah. There were people in JTS, however, who were clearly Frum, believing people. Most notably, for example, R’ Shaul Lieberman. He was undoubtedly Kosher and any Geirus under his auspices would be fine. The same is true of several of his Talmidim, who came through JTS totally Frum. They were rare, certainly, and by now are possibly nonexistent, but they did exist.
About Ethiopians: It is a massive Machlokes HaPoskim whether they are Jewish or need Geirus. In the end of the day, everyone seems to hold like R’ Ovadia that we treat them as full-fledged non-Jews until they convert. This isn’t because we don’t believe their history; rather, it’s because if we treated them as Jews (or even Safek Jews), the standard Psak would have to be that we would treat them all as Safek Mamzerim.