I am surprised that not one poster mentioned the origin of the psak that Ethiopians Jews are indeed jews. It came from the Radbaz- the main Possek of his time (about five hundred years ago) and predated the Bais Josef. He “paskened’ that they were descendants of Shevet Dan and R,Ovadia Yossef zz’l did indeed rule accordingly that Ethiopian jews are full jews.
As far as mamzeirus goes, there is no such thing as a “sofek mamzer”. One is either a “mamzer”(if it can be proven) or a jew. The gemoro clearly says “mamzer vadai omar rachmono velo mamzer soffek”.