Are the elements of an averaging kernel matrix always centered on the diagonal, or can they be exclusively off-diagonal in rare cases?
This statement from Geophysical Data Analysis: Discrete Inverse Theory: William Menke should make things clear for you


  The diagonal elements of the covariance matrix are a measure of the
  width of the distribution of the data, and the off-diagonal elements
  indicate the degree to which pairs of data are correlated.


Regarding the second part of your doubt whether a situation is possible where ds = 0 ; this would imply some error in measurement(for example - some noise) and would not have any implication on the geophysical nature of the data . It would be advisable to take multiple readings at different locations in this kind of situation.

Finally ds provides some insight into the quality of data collected but no true information about the "information content of measurement".