“Lines p and q determine one plane beta.
Then planes alpha and beta have one common point A (A belongs to line q).
Then there is common line q' for alpha and beta.
Line p is parallel to line q' (otherwise it would intersect plane alpha) but then is q'=q because only one line is parallel to line p which goes through point A, so q belongs to plane alpha.”
It seems to me as if there is problem with “Line p is parallel to line q' (otherwise it would intersect plane alpha) but then is q'=q”. Just because two lines do not intersect does not make them parallel.
Skew lines have that property.
You can say that both p & q’ are in plane beta. Because q’ is a subset of alpha it cannot intersect p. Therefore in plane beta we have two non-intersecting lines so they must be parallel.
Now the rest of your proof works.