Genetically speaking, who are the pre-Islamic Arab tribes who lived in Yemen and Arabia in many centuries B.C. closest to? Where did the first Arab settlers in Yemen come from? I know that Arabs claim to be descendants of Abraham, is their any proof to that claim, and if so, how did they come to live in Yemen and Arabia?
There is little reason to believe in the historicity of Abraham. He could be a real person, a character composite of many different people, a symbolic representation, or just entirely fictional. And accordingly, there wouldn't be any proof as such that anyone is descended from him. Even if archaeologists were to miraculously stumble upon his remains, it is doubtful whether we could ascertain his identity.
The earliest attested inhabitants of Yemen were several Ṣayhadic tribes: Sabaeans, Hadhramis, Qatabans, and Minaeans. These Old Southern Arabic peoples formed from South Semites who may have reached Yemen some time around 2000 B.C., and probably came from somewhere around the Fertile Crescent. Here the Akkadian Empire (in Mesopotamia) and the Amorite Kingdom (in Syria) emerged around 2500-3500 B.C. as the earliest Semitic polities. And the Semitic peoples in general seemed to have originated somewhere in the Near East.
Also, the precursors to the first Semites probably reached the Levant from the Afro-Asiatic homeland around the Nile, in Africa.
Genealogy is family history based on records. There exists no such records that far back in time, so the question is not answerable. If we widen the question more, some sort of answer is possible however:
People having been living in Arabia since the first humans migrated out of Africa, and as pretty much everywhere in the world, they have moved in and out of the area and mixed with neighbors.