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After much looking around I eventually found a partial answer in a few different sources, but primarily in samples/cpp/tutorial_code/calib3d/camera_calibration/in_VID5.xml . In the sample configuartion file it has:

  <!-- Number of inner corners per a item row and column. (square, circle) -->
  <BoardSize_Width> 9</BoardSize_Width>
  <BoardSize_Height>6</BoardSize_Height>

So this answers one of the questions: 9x6 represent the inner corners.

Finally, it has:

  <!-- The size of a square in some user defined metric system (pixel, millimeter)-->
  <Square_Size>50</Square_Size>

Though this still doesn't fully answer the DPI question since it's not clear where we define the metric system (does anyone know)?

However, from what I have read, the real size of the printed squares does not matter for the calibration coefficients, but instead only matters for the extrinsic values used in calculating the pose. Therefore, for the purposes of simple calibration, it doesn't matter.