Asia Minor was the Ancient Greek term for it. I don't think it's actively used these days, other than in historical accounts; at least not in English. Anatolia is the more common word for that region.
Ural mountains are the eastern boundary of Europe, not the northern one (they stretch from north to south). The northern boundary is the Arctic ocean. Black Sea is the southern boundary, along with the straights, and further east it's Caucasus mountains and then Caspian sea.
However, this boundary is not only arbitrary, it's relatively (as in, only a couple of centuries) new. There were other definitions before, with various major rivers used as the eastern boundary (e.g. Don or Dnieper). This is because there really isn't any good geographic definition, nor is there a particular need for one - the continent is a single one, Eurasia. Historically, Europe has been more of a political division than geographic (geography played into it only to the extent of defining easy to protect natural boundaries, like rivers and mountains, which then tend to become state boundaries.).
Afghans are actually not Arabs at all (and Arabs are themselves Semitic people). The majority of Afghans are Pashtun, which is an Iranian sub-ethnicity, and the language that they mostly speak is Pashto, an Indo-Iranian language closely related to Persian; the second most popular language is Dari, which is a dialect of Persian. The second largest group are Tajiks, which are also an Iranian sub-ethnicity and speak an Indo-Iranian language very close to Persian. Then come Hazara, who speak Dari and are also Indo-Iranian; Uzbeks, which speak a Turkic language; and Balochi, who are again Indo-Iranian. So none of these are Arabic or Semitic in any way. Arabs and Arab-speakers are a tiny minority in Afghanistan (which kinda makes sense when you realize that it's basically all former territories of Persian Empire dating all the way back to Achaemenids, so ethnically and linguistically it's rooted in Iranian culture.
Also, just FYI, in contemporary American usage, "Oriental" is seen at best an archaism deliberately used to evoke the atmosphere of the times when it was heavily used, and at worst is actually considered derogatory (or culturally alienating; either way, carrying a distinctive implication of racism).