Discrete Cosine Transform (DCT) Coefficient Distribution
Probably the value of the DCT coefficient and its frequency (i.e., a histogram plot). However, it's impossible to say that with certainty given the information you have provided. If you found that plot in the linked paper, then the definitions of the x- and y-axes are in all likelihood given therein. The paper is behind a paywall, so it's not helpful to anyone without an IEEE account.
Yes. Check the scipy documentation for the dct() function.
See 1.
This is impossible to answer. What does appropriate mean in this context?
To classify a coefficient as either low, middle, or high frequency, compute its distance from the DC bin. For bin (m,n), its distance will be r = sqrt( (m-1)^2 + (n-1)^2 ). Judging by the picture you included, it looks like low frequencies are those with r &lt; M/2, middle frequencies are those with M/2 &lt;= r &lt; M, and high frequencies are those with r &gt;= M.


EDIT:

From the linked paper:


  Furthermore, it has been explained that noise addition to an image
  changes its DCT coefficient distribution [9], as we confirm in Fig. 2
  for a document image. Part (a) of the figure is an example of
  noiseless document image, part (b) is the noisy version of image (a)
  due to Gaussian blurring and Gaussian noise addition, and part (c)
  shows the higher frequency DCT coefficient distribution of of part(a)
  and (b) respectively. A lower-peaked histogram belongs to the
  ground-truth (GT) image, while a sharp-peaked histogram belongs to the
  noisy document image.


So part (c), which is the image you included, shows the histograms of the "higher frequency" DCT coefficients. Thus, the x-axis is the value of the DCT coefficient for the high frequency band, and the y-axis is its normalized histogram value. (The normalized histogram value is sometimes called "frequency", but that can get confusing in this case since DCT coefficients also represent frequencies. The normalized histogram value is just the number of occurrences of a certain value divided by the total number input samples to the histogram.)