Align Objects to a Line with Different Heights

If I understand your request, the script below should get you the desired result (see pic below). I tried to trace your reference path as close as possible for testing. I have also included before and after vector files for you to work with as well.

Looking at your example picture, the path starts and ends on a horizontal segment. With this in mind, the script just iterates over each set of 2 points, determines the center point between, and then draws a cross mark (offset above the path). All cross marks are put in a separate layer and grouped.

I set everything up to be customized, cross color, cross size, and cross offset. See comments in script...

Try the script out and let me know if it does what you need.

(function () {
  // setup variables
  var doc,
    sel,
    selectedPath,
    layer,
    crossMarksGroup,
    crossMarkSize,
    crossMarkVerticalOffset,
    crossMarkColor,
    p1,
    p2,
    centerPointX;

  // no need to continue if there is no active document
  if (!app.documents.length) {
    alert("No active document.");
    return;
  }
  doc = app.activeDocument;

  // no need to continue if there is no active selection
  if (!doc.selection.length) {
    alert("No active selection.");
    return;
  }
  sel = doc.selection;

  // get the **FIRST** pathItem of the selected objects
  for (var i = 0; i < sel.length; i++) {
    if (sel[i].typename === "PathItem") {
      selectedPath = sel[i];
      break;
    }
  }

  // no need to continue if no pathItems were selected
  if (typeof selectedPath === "undefined") {
    alert("No paths were selected.");
    return;
  }

  // create a new layer to hold the lines (use old if already present)
  try {
    layer = doc.layers.getByName("SCRIPT CROSSES");
  } catch (error) {
    $.writeln("layer not found");
  }
  if (typeof layer === "undefined") {
    layer = doc.layers.add();
    layer.name = "SCRIPT CROSSES";
  }

  // create a group to hold all cross marks
  crossMarksGroup = layer.groupItems.add();

  // setup cross mark
  crossMarkSize = 9; // points
  crossMarkVerticalOffset = 18; // point above line
  var crossMarkColor = new RGBColor();
  crossMarkColor.red = 255;
  crossMarkColor.green = crossMarkColor.blue = 0;

  // iterate over each set of 2 points and draw cross marks
  for (var i = 0; i < selectedPath.pathPoints.length - 1; i += 2) {
    p1 = selectedPath.pathPoints[i];
    p2 = selectedPath.pathPoints[i + 1];

    // skip set if y-values don't match
    if (p1.anchor[1] != p2.anchor[1]) {
      continue;
    }

    // calculate center point between x-values
    centerPointX = p1.anchor[0] + (p2.anchor[0] - p1.anchor[0]) / 2;

    // draw cross at center point
    drawCrossMark(centerPointX, p1.anchor[1]);
  }

  function drawCrossMark(x, y) {
    // make a group to hold cross mark parts
    var crossGroup = crossMarksGroup.groupItems.add();
    // draw x-line part
    var xLine = crossGroup.pathItems.add();
    xLine.setEntirePath([
      [x - crossMarkSize / 2, y + crossMarkVerticalOffset],
      [x + crossMarkSize / 2, y + crossMarkVerticalOffset],
    ]);
    // draw y-line part
    var yLine = crossGroup.pathItems.add();
    yLine.setEntirePath([
      [x, y + crossMarkVerticalOffset + crossMarkSize / 2],
      [x, y + crossMarkVerticalOffset - crossMarkSize / 2],
    ]);
    // set cross mark styling
    xLine.strokeColor = yLine.strokeColor = crossMarkColor;
    xLine.stroked = yLine.stroked = true;
    xLine.strokeWidth = yLine.strokeWidth = selectedPath.strokeWidth;
    xLine.filled = yLine.filled = false;
  }
})();