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No, only sane people question their sanity. The fear of going crazy is a symptom to many people who suffer from depression, anxiety and panic disorders. Those who actually suffer from insanity don't realize it.
No but I think constant fear can lead to some unhealthy mental coping skills. Avoidance, hypervigiliance, manipulation, repression of feeling and all kinds of stuff. The end result would not be crazy just maladjusted to the world around and that can feel crazy.