No, because with schizophrenia things that should have no salience whatsoever become very salient to you. I.e. scales all being rectangular (thus making the diet industry run by a bunch of Pythagorean rectangle-worshipping cultists) or something.

No. Having schizophrenia may result in you not being able to distinguish between reality and dreams when you are awake.

While you are asleep, it's quite common to believe in your dreamscape. (It's also common to tell yourself that it's a dream.)