Could a sociopath have a breakdown and be hospitalized for it?

Yes. Sociopaths can and often have other conditions like bipolarity that may lead to 'breakdowns'. The will get depressed often after being 'caught' or exposed for their misdeeds. Depressions are often short-lived and not deeply felt or experienced given the nature of their sociopathy.
Possibly. It is possible, if life circumstances are not to the person's tolerance level. Generally people with such personality issues do not come for mental health care unless forced to by others in their lives, or are in prison systems where behavior leads to psychiatric examination. People who truly have antisocial personality disorder usually cause more pain in others than they feel themselves.
Possibly I suppose. This is possible, I suppose, depending on how you define what a breakdown is. Having said that, most sociopaths (AKA antisocial personality disorder) people wind up in legal troubles or are incarcerated, as opposed to having a nervous breakdown and having to be hospitalized.