Given two noncoplanar lines p and q, and a point A, does there always exist...

Step 1
Once you have the plane $\alpha (p,A)$ , this happens if q belongs to a plane which is parallel to $\alpha$ .
For example, let p be the x axis and $A=(0,1,0)$ : the plane containing them is $z=0$ . Any line passing through p and A is confined on this plane (axiom I.6: "the line lies on the plane"). Now imagine q being on the plane $z=1$ and you have your counterexample.