A water softener uses a cation exchange resin to remove hardness; hardness is comprised of dissolved "positive" metal ions such as calcium, magnesium and, to a lesser extent, iron. Chlorine is a negatively charge oxidizing ion. The resins used in water softeners are only capable of removing positively charged ions dissolved in water. They are not capable of removing chlorine. To remove chlorine you need to use activated carbon.
Sprite Shower filters does not remove chloramines.
No. If anything, it will would make your bathwater harder. To remove hardness, you need a cation exchange resin, such as that found in a water softener. The salt consumed in a water softener is used to recharge the exchange resin so it can capture the dissolved metal ions (calcium and magnesium.) This is simply a blend of salts that are added directly to bathwater and it will not remove hardness.