About your iron relics (photo #1):
Upper left, about half of ONE SIDE of a horse-bridle "snaffle bit." See the photo below to view a fully intact one. Unfortunately, a snaffle bit cannot be dated with 100% certainty because snaffle bits have been being manufactured from before the civil war all the way up to today. And, although a few million horses were used by the armies during the civil war, over 100 million horses were used by civilians from 1866 into the 1930s. That's why most of the horse-harness buckles & bits, etc that we dig was civilian-usage stuff and was from after the civil war.
Upper right, a miscellaneous iron rectangular buckle. As with your snaffle bit, this buckle being dug in a yankee camp, it's probably military, but no specific ID is possible... it could be from the bottom of a cartridge-box, or a soldier's knapsack strap, or horse-harness, or quite a few other leather-strap possibilities.
Lower left, part of a buggy or wagon seat leaf-spring assembly.
Lower right, Beshires 1 is correct, it is a horseharness (or mule harness) back-band hook. More specifically, it appears to be a "Plow Boy" back band hook, from plow-harness. See the diagram below, scanned from a very-early-1900s saddlery & horseharness catalog.