R1a/b are Iranian-speaking steppe tribes (literally Aryans) who raped their way across all of Eurasia. In Europe R1b became associated mostly with Celts/Germanics (forest dwellers) and R1a with proto-slavs (steppe dwellers). J originated in the Caucasus mountains (thus literally Caucasian) and made it's way across the mid-east and the Mediterranean coast and into the Arabian peninsula (it did not originated there). This occurred before R entered Europe, and before "Arab" or "Semite" was a meaningful term. J1 is primarily associated with the Levant and Arabian peninsula, J2 with Mediterranean Europe, Anatolia, and the Caucasus. J1/2 were mingled in Mesopotamia.
J1 re-entered J2 "territory" with the Arab Muslim conquests. E spread around the Mediterranean basin 10s of thousands of years ago, and is today primarily associated with berbers. E3b has been present in Europe for longer than R1a/b. Though the E family originated in Ethiopia, this was iirc 80,000 years ago. The specific clade of E found in Europe is distinct from that which remained in Ethiopia (all haplogroups having originated in Africa, this is largely irrelevant). Berber, Mediterranean European, and Ethiopian clades of E are all distinct. Iirc, I1a/b is associated with the Paleolithic pre-agricultural population, which was mostly exterminated by R1 after I1 had mingled with J2 farmers who migrated from Anatolia.
Long story short, J2 is pretty much what makes Mediterraneans different from northern Europeans, it originated in the Caucasus (as did R1a/b) and though present at high levels throughout the middle East, is in that region mixed with large numbers of haplogroups that are not present in Europe, north or south.
Also, this only gives us half of the picture (patrilineal ydna), because mtdna is far more difficult to map onto historical events due to roasties being traded like currency throughout the ages.