what exactly do the dotted lines represent in this diagram?
The dotted lines are possible partition points of the X axis (values along X) that would minimize the surface area heuristic. Notice that they correspond to the start-end of primitives from the perspective of this axis. Keep in mind, the idea is to assign triangles to partitions based on whether their centroid lies to the left or the right of the partitioning plane.
A partition point somewhere in the middle of a primitive wouldn't make much sense, since the bounding box of the primitives to the left and right of the partition would still encompass the bounding box of the primitives.
In short: the algorithm is: for all points marked by dotted lines, split primitives based on whether their centroid falls to the left and right of the line, then pick the dotted line for which the SAH is minimized.
what metric do we use to determine which partition is better?
We need to consider even more larger number of possible planes in 3D...
What does the horizontal arrow line represent in this diagram?