A short answer for you, the sling was popular around the world and not only by pastoralists nor Levantine people. The earliest evidence of slings is in a mural at Catalhoyuk (ca. 7000 BCE), with the earliest archeological evidence being a Norte Chico culture (ca. 2500 BCE) sling at the National Museum of Anthropology & Archaeology in Lima, Peru; and in late bronze age Afro-Eurasia there's a sling in Tutankhamun's grave and sling shot found (along with arrows and other weapons) in the destruction layers of Hisarlik (Troy). The simplicity of the weapon, along with its ubiquity around the ancient world (along with a similar weapon, the bola), suggest that humans have likely been using this weapon since the paleolithic. I cannot verify further details given by Chris Harrison, who cites historians who give a detailed history of the neolithic and chalcolithic use of the sling...
Near Eastern armies began supplying their slingers with uniform projectiles, made from baked-clay or carved stone, by the end of the 7th millennium B.C.. At first, these were spherical, but by 3000 B.C., biconical or ovoid projectiles were discovered to be superior. The latter two types would orient point first and spin through the air like a bullet or American football (Hawkins, 1847; Korfmann, 1973; Ferrill, 1985; Carman, 1999).
But your second question is more difficult - how were these skills passed down? Well, the sling is easy to use but difficult to master; and all ancient authors agree that the greatest slingers were trained since they were young children. Just as children learned archery and weaving from their parents starting when they were only a few years old, presumably they also learned how to use a sling. But exactly how this was done, no one thought it necessary to write down (sadly, this is the case for so many fascinating aspects of social history). Strabo in his Geography (Book 3, Chapter 5) is one of the first to mention this:
The long sling [Balearic people] use for hitting at far distances, the short one for near marks, and the middle one for those between. From childhood they were so thoroughly practiced in the use of slings, that bread was never distributed to the children till they had won it by the sling.
The Roman writer Vegetius says about the same, ...owing to the manner of bringing up their children.
Recruits are to be taught the art of throwing stones both with the hand and sling. The inhabitants of the Balearic Islands are said to have been the inventors of slings, and to have managed them with surprising dexterity, owing to the manner of bringing up their children. The children were not allowed to have their food by their mothers till they had first struck it with their sling. Soldiers, notwithstanding their defensive armor, are often more annoyed by the round stones from the sling than by all the arrows of the enemy. Stones kill without mangling the body, and the contusion is mortal without loss of blood. It is universally known the ancients employed slingers in all their engagements. There is the greater reason for instructing all troops, without exception, in this exercise, as the sling cannot be reckoned any incumbrance, and often is of the greatest service, especially when they are obliged to engage in stony places, to defend a mountain or an eminence, or to repulse an enemy at the attack of a castle or city.
Ancient authors also all agree that expert slingers could hit incredibly precise targets, so if you had been raised doing this since childhood then you could be very good. But how did people get so proficient, perhaps they were instructed by parents or through hunting small animals...while I think it's probably the latter we simply don't know the details.
The Greek Age of Bronze: Trojan War, by Andrea Salimbeti http://www.salimbeti.com/micenei/war.htm
De Re Militari, Book I: The Selection and Training of New Levies, by Vegetius
