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I have written a blog post explaining the effect of the axis skew on the calibration matrix.

In brief the answers to your questions are

Q-1 How coordinate system is transformed when there is skew?
Answer: Using the parallelogram of vector addition and some trigonometry, you can derive the skew coordinates in terms of the cartesian coordinates of a point

Q-2 How pixel scaling is done?
Answer: It depends on the transformation you choose. Some prefer shearing (which changes the pixel height) and others prefer a pixel dimension preserving transformation.