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To answer your first and third questions, I would recommend doing a parametric sweep on the distance between the feed point and the edge of the patch. I'm not familiar with CST, but the best point is of course where the port impedance matches your system impedance (50 Ohms?). Alternatively, if you have the scattering parameters, you want the point which gives you the lowest S11.
For an analytical solution to all of your questions, I would recommend Balanis' antenna book.
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