The Greek mercenaries were hired by the Persians as shock infantry-- one of the units in which the Achaimenid army was deficient. Slingers and archers, the Persians had in great numbers. It's part of a passage where the Greeks, stranded in Mesopotamia, have to invent all kinds of expedients to make themselves into a multi-functional army: they also improvise cavalry. Yes, the Greek command buys slings off people who have them, and pay others to braid slings. They used sinews they found in villages, and also took lead and cast bullets.

There are two schools of thought on this. One, by Pierre Briant, argues that this stuff is from strategic stores of the Achaimenid army, along with bowstrings, which the Greeks plaited into slings. The problem is that lead bullets are not attested for the Persian army-- that's specifically what Xenophon says. (Of course, sling bullets do turn up in Tissaphernes' name; the argument then would be that the Persians adopted sling bullets after fighting against Greeks--Tissaphernes was the western satrap with lots of experience of fighting Greeks, and specifically led cav. against the Ten thousand). See P. Briant, History of the Persian Empire, on this.

The other view is that these are not bowstrings, but sinews; not lead for bullets, but artisanal. Picked up in a number of villages. Is known as one of the metals for artisanal stuff. Not used for roofing and guttering (I would guess Mesopotamian houses have flat roofs sealed with pitch, or reeds), but for vessels, often very large.

(the passage has been discussed a few times before). They sling large stones (hand-filling stones). They are local forces under a local commander, so perhaps Iranian levies or levies from modern N. Iraq-- this is not the invasion of Greece ca. 480.

They were all heavy infantry, the cavalry units and slingers were created out of them, multi-tasking army. There also are light infantry in Cyrus' army which are not Greek, and which must have tagged along when the Ten Thou marched west.

May I recommend the Anabasis to everyone as one of the most interesting texts out of the ancient world ? There is a volume of essays on it R. Lane Fox, The Long March: Xenophon and the Ten Thousand, which is quite good.