Is it possible to get the locations and dimensions of all boxes in a StandardForm expression renderation?

This is not yet a complete answer but I think it may set you on a viable path. We can style each element using StyleBox as follows:

boxes = RowBox[{"(", 
    FractionBox[
     RowBox[{RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], SuperscriptBox["c", "d"]}], 
     SqrtBox["e"]], ")"}];

colors = ColorData[54, "ColorList"]

boxesNew =
  Module[{i = 1},
    boxes /. s_String :>
      (StyleBox[s, FontColor -> #, Background -> #] & @ colors[[i++]])
  ];

The result:

boxesNew // DisplayForm

The object selected, revealing its parts:

We can create an Image with Rasterize:

img = Rasterize[boxesNew // DisplayForm, "Image"];

This image could then be processed to identify each area of solid color and these mapped back to the color list in colors. (The colors chosen are only for human readability; sequential RGB values might be used for many segments.)

Details depend of your specific needs but it seems that information you desire is at least partially accesible via FrontEnd`UndocumentedGetSelectionPacket.

RowBox[{"(", 
   FractionBox[
    RowBox[{RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], 
      SuperscriptBox["c", "d"]}], SqrtBox["e"]], ")"}
] // RawBoxes

Dynamic[
 FrontEndExecute @ FrontEnd`UndocumentedGetSelectionPacket[EvaluationNotebook[]], 
 UpdateInterval -> 1
]

VirtualSelectionRectangle seems to give {xmin, xmax}, {ymin, ymax} where x are pixel columns counted from left edge of the notebook and y coordinate determines pixel row counting from top edge (NOT from the beginning of the notebook).