Do patients with wilson's disease face any particular social stigma?

Not from diagnosis. It is unlikely that the diagnosis itself of wilson's disease (inherited tendency to accumulate and deposit copper in organs including liver and brain) carries a social stigma, but the psychiatric disturbance and neurologic signs that can accompany wilson's unfortunately may. I am hopeful that our society is moving towards the point at which psychiatric illness no longer carries a stigma.
Not to my knowledge. Some of the other genetic diseases are targets of superstition in some cultures -- infamously thal minor among some Greek communities.
No. There is no social stigma for wilson's disease, but some patients have neuropsychological issues that create social problems for them.