Adds Hardness to the Bar?:
Most definitely! Due to it's high stearic and palmitic fatty acid content, even at small amounts (5-10%), cocoa butter adds hardness to the final soap bar.
Creates Fluffy Lather?:
No... In fact, too much of any of the butters will really kill fluffy lather.
Creates Stable Lather?:
Yes...the lather is low and creamy and stable.
Adds Moisturizing Qualities?:
Most definitely! Cocoa butter, and all of the nut butters are GREAT providers of skin-loving qualities in your soap. Very moisturizing in soaps or just straight out of the jar.
Cocoa butter, even though it's more expensive than other base oils, is one of the staples in my basic soap recipe. Even at 5% or so, it imparts lovely moisturizing and helps create a nice hard bar. Natural cocoa butter has a yummy warm chocolate scent that will come through in balms and lotions, and even comes through in soap if you use enough in the recipe and don't mask it with a strong fragrance oil. The deodorized/refined cocoa butter has much less scent.