Displacement results always converge at a lower node count than stress results because stress is derived from displacements. The difference in displacement of adjacent nodes is used to compute element stress. Since each node has its own displacement error, the errors combine in the stress result, leading to a larger element stress error.

Are those nodal forces that you use on the flank?  A nodal force on the corner nodes of a quadratic element while midside nodes have zero force does not deliver a uniform pressure profile, but an oscillating pressure profile. I suggest you switch to PLANE182 elements to avoid this problem and use smaller elements.

Does the node on which the peak stress lands have a force on it or not?

Is the Stress Averaged or Unaveraged?

It is better to plot the element size of the elements in the vicinity of the peak stress rather than the node count of the whole model. Due to the way meshing is done, the element size in the vicinity of the peak stress might not change in proporortion to the total node count.