Is it true that a phycopath is not mentally ill, nor physically ill or emotionally ill, but in fact it's that they've chosen to be the way they are?

Not true: The term psychopath or sociopath are used in different ways and are not diagnostic terms. They are often used to describe people who others believe behaved negatively, manipulatively, or harmfully without remorse. People tend to fall on a wide spectrum regarding any label or personality trait, including the ones you are referring to. As used originally, a psychopath was mentally ill.