Yes. Waxy flexibility is a symptom that can commonly occur in the context of catatonia but is not a required symptoms to diagnosis the condition.
Yes. Catatonia comes in 2 forms -- the waxy flexibility, non-moving mode and also a frenzied movement mode. Psychiatric nomenclature is moving away from categorizing different forms of schizophrenia, though. Plus catatonia does not automatically mean schizophrenia.
Catatonia. Waxing flexibility isn't a requirement to have catatonic schizophrenia. Most catatonic cases that I have seen have involved episodes of immobility, actually (ie, freezing in place/position). Of course, the person has to meet other criteria for schizophrenia, too--the catatonic state by itself isn't enough for the diagnosis.