Is there a connection between narcissism and schizophrenia?

No. Schizophrenia is an axis 1 thought disorder characterized by psychosis. Narcissism is an axis 2 cluster b personality trait. It is possible to have one, the other one, both or neither.
No. Narcissism is a personality disorder, featuring grandiosity, egomania, devaluation and objectification of others, feelings of specialness and entitlement. Schizophrenia is one of the psychotic disorders, featuring (typically) auditory and visual hallucinations (and less frequently tactile, gustatory, and olfactory hallucinations.
Possibly. There are a number of possible connections. The most important one is that any chronic disease (such as schizophrenia) is likely to make you more preoccupied with yourself (and thus make you seem narcissistic). In addition, schizophrenia, which is primarily a disorder of unusual brain connections, makes people think in ways that are very personal and don't make sense to others.