sharpness
No, I think there is no unit for measuring sharpness.
No, the word sharpness is a common, abstract, uncountable noun, a word for the quality of a thing.I can't think of a use for the noun sharpness in the possessive form; I don't know what the quality of sharpness could posses. If there is a use for the possessive form, that would be: sharpness's.
One can improve the sharpness of his mind by maditation and yoga.
Sharpness Branch Line was created in 1876.
Sharpness railway station ended in 1964.
Sharpness railway station was created in 1876.
Resolution is another name for image sharpness.
The noun is sharpness.
Usually sharpness of a monitor is referred to as its Dot Pitch.
The sharpness of vision is known as visual acuity. Source: psychology class
It adds more damage. You'll see a couple particles flying from your sword when you hit the opposing mob - that is the sharpness kicking in. The higher the sharpness level, the higher the damage.