1861 Ordnance Manual requirements stated the following Infantry accoutrements: Cartridge box, cartridge box plate, cartride box belt, cartridge box belt plate, cap pouch, bayonet scabbard, waist belt, waist belt plate & Gun Sling.
Accoutrements were shiped seperately from the "stand of arms" which included the long arm & bayonet. One of the arguments in re-enacting today is just how common the Enfield bayonet scabbard really was, by study it appears to be a modern re-enactorism to see so many in the ranks. The most recent research shows they should only be worn w/ English manufactured accoutrements and frankly more common in CS useage.
Of note on slings, the majority of slings used by the US were of the same length as used for the Springfield M1842 & M1855/61 series of arms. Thus they were a touch short for the Enfield but still worked well enough. There is argument as to if the Brits sold the CS any slings for the P53 series arms, though there is evidence some were on blockade runners so at least some English made slings for Enfields made it into CS service add to that the CS did manufacture of canvas/leather combo slings.
The P53/56/58/61 were all designed to be slung muzzle down while US manufactured arms were the opposite.
That said I have seen quite a few original CDV's of Union soldiers w/ slings and probably as many w/out.
Also keep in mind at the time of the CW the sling was not intended to be used to steady the weapon as became common later on.