Leaflet custom coordinates on image

If I understand correctly, you want a CRS similar to L.CRS.Simple that places tile 0/0/0 (tile size 268px, which is 8576 / 2⁵) so that:

• Position [0, 0] is at the center of that tile.
• The entire world (i.e. entire tile 0/0/0) goes from position [-8576/2, -8576/2] to [8576/2, 8576/2].

You would just need to adjust the L.CRS.Simple with the appropriate transformation, to account for this scale of 1/2⁵ = 1/32 (instead of just 1) and offset of 8576 * 1/32 / 2 = 268 / 2 = 134 (instead of 0.5).

L.CRS.MySimple = L.extend({}, L.CRS.Simple, {
  transformation: new L.Transformation(1 / 32, 134, -1 / 32, 134)
});

var map = L.map('map', {
  maxZoom: mapMaxZoom,
  minZoom: mapMinZoom,
  crs: L.CRS.MySimple
}).setView([0, 0], mapMaxZoom);

Demo: http://plnkr.co/edit/5SQqp7SP4nf8muPM5iso?p=preview (I used Plunker instead of jsfiddle because you provided a full page code with HTML, whereas jsfiddle expects you to split your HTML, CSS and JavaScript codes into separate blocks).

As for showing the coordinates and a "locate" button, it would be quite easy to implement so that it is similar to the example you mention. Feel free to open new questions if you need help.

In the above demo, I used Leaflet.Coordinates plugin to implement quickly both functionalities (see the control on bottom left corner of the map; you have to start moving your mouse on the map for the coordinates to appear; click on that control to open the edition mode).

EDIT:

As for the Leaflet.Coordinates plugin, it wraps displayed coordinates longitude to stay within [-180; 180] degrees.

In your case where coordinates are not degrees, there is no point wrapping the longitude.

I think this is the cause for the discrepancy of coordinates between the click popup and the control.

Simply patch the plugin code to prevent wrapping:

// Patch first to avoid longitude wrapping.
L.Control.Coordinates.include({
  _update: function(evt) {
    var pos = evt.latlng,
      opts = this.options;
    if (pos) {
      //pos = pos.wrap(); // Remove that instruction.
      this._currentPos = pos;
      this._inputY.value = L.NumberFormatter.round(pos.lat, opts.decimals, opts.decimalSeperator);
      this._inputX.value = L.NumberFormatter.round(pos.lng, opts.decimals, opts.decimalSeperator);
      this._label.innerHTML = this._createCoordinateLabel(pos);
    }
  }
});

Updated demo: http://plnkr.co/edit/M3Ru0xqn6AxAaSb4kIJU?p=preview