While the origin of the weapon is lost to time, slings were commonly used throughout the Ancient Mediterranean, from Iberia to the Near East. They had a reputation for being a bit of a hillbilly weapon - stereotypically used by shepherds to protect their flocks from wolves - but this didn't stop major powers from raising large numbers of slingers for their campaigns. Evidence of the use of slings is found in Roman legionary encampments and in the siege sites of Philip II of Macedon. Entire regions, such as the Balearic islands, Achaia in the Peloponnese, and Rhodes, were famous as recruitment grounds for capable mercenary slingers.

The sling had several advantages as a weapon. First, while it was not easy to master its use, it was at least cheaper than a bow and arrow; the weapon itself was just a strip of cloth, and small stones could serve as ammunition. Shaped lead bullets were a much more effective projectile, but even these could be quickly and easily made as long as lead was available. Second, as the fourth-century BC Greek author Xenophon attests, practiced slingers easily outranged archers, making them ideal to protect a marching column against enemy missile troops. Third, the penetrative power of sling bullets made them effective even against armoured targets, and very useful to clear city walls of defenders.

I'm no expert on Medieval warfare, but to my knowledge the use of the sling never really fell out of favour; slingers simply weren't recruited as specialist light-armed warriors anymore. Some regions nevertheless continued to hone their skill at the craft, and allegedly there were Spanish soldiers using slings to throw handgrenades as recently as the Spanish Civil War.

Due to their high velocity, sling projectiles were no toys, and I doubt it would have been easy to make sling bullets of a material that could hold together in flight but not hurt its target. I don't know of any precedent for that. Xenophon (again) describes a fictional mock battle between missile troops and melee troops in the army of Cyrus the Great, but in this exercise the latter are armed with sticks and the former simply throw clods of mud.