1. With the field given by (z - x²)i - yj + sin(x)k, where are there sources and where sinks?

Since the divergence is 1 - 2x, it is positive (points are sources of arrows) when x < 1/2
and negative (all points are sinks) when x > 1/2.

2. Where are sources and sinks for the field r^-3?

This field has a source at the origin and arrows point radially outward; there are no finite sinks; the arrows,  if connected into lines, all go out to infinity.
Notice that the origin is definitely a source of arrows.