So f cannot be differentiable at any point except possibly the origin. This shows that f is not analytic anywhere, but does not preclude the possibility that it is differentiable at a few isolated points. So f is not analytic at the origin. Observing that f was differentiable with respect to x and y, and then concluding that f must be differentiable with respect to z. Assuming that, since f is differentiable at the origin, that f must also be analytic at the origin.
Analyticity is a stricter property than differentiability.
These are all phases of -i, but they are not the principal phase. This is not how principal branches work. A little experimentation, though, will give other branches which will work. Again, this is not what the question asked. Read the question carefully.