Why is cell constant value measured through indirect methods (like using a solution whose conductivity is already known)? Isn't it easier to measure the length and area of cross section of a conductivity cell directly?
The mathematical description of the cell constant ($l/A$) does not include factors like fringing fields (the ionic current does not travel precisely in a well-defined path between the electrodes) and the actual surface area for rough surfaces.
The cross section of a round electrode may have different fringing fields depending on its separation from its companion. And if the two electrodes are of different geometry, you have to decide how to average the areas out. I think there was more standardization in the choice of electrode materials, size and spacing than has been carried over into the beautifully simple formula $l/A$, so when the formula is used outside of certain narrow bounds (and I don't know what they are), it's only an approximation, and resistance or conductance can be measured more precisely than the cell constant from geometry. So you might as well calibrate against a known cell or solution.
The conductivity value of any cell changes with the area of cross-section of the electrode. When we use an electrolyte of concentration within the desired area, the conductivity value varies. When a standard value of specific conductance is given, then from the observed conductance value we can calculate the cell constant.