Good meaningful questions which get to the 'heart of the matter'.
All cuts on the potato should be exposed to sun to callous over (scab on the cuts).
I like to include at least a 2 by 2 inch piece of the original potato along with the 'eye', and thuroughly calloused by exposure to sunlight.
Potatos are a member of the solanacious family (solar = sun light) as are peppers , tomato , eggplant , etc.
Crop rotation should be practiced so as not to plant solancious plants where other had been planted in the last 3 - 4 years; otherwise there may be pest/disease in the soil.
Yeah, 2X2 is good. I usually figure on at least the size of a small egg, and a minimum of 2 eyes.