Leaflet custom coordinates on image

I have an image which size is 8576x8576px, and I want to make the coordinates match 1:1. Also I want the coordinates 0,0 in the center of the image (now the center is -128,128). And I want to show the coordinates too. I want to put a locate button for the user insert coordinates and then find them on the map. Something like this: http://xero-hurtworld.com/map_steam.php (I am using the same image but bigger). The tile size I made its 268px.

My code so far:

https://jsfiddle.net/ze62dte0/

<!DOCTYPE html>
<html>
  <head>
    <title>Map</title>
    <meta charset="utf-8"/>
    <meta name="viewport" content="initial-scale=1.0, user-scalable=no"/>
    <link rel="stylesheet" href="http://cdn.leafletjs.com/leaflet-0.6.4/leaflet.css" />
    <!--[if lte IE 8]>
    <link rel="stylesheet" href="http://cdn.leafletjs.com/leaflet-0.6.4/leaflet.ie.css" />
    <![endif]-->
    <script src="http://cdn.leafletjs.com/leaflet-0.6.4/leaflet.js" charset="utf-8"></script>
    <script>
      function init() {
        var mapMinZoom = 0;
        var mapMaxZoom = 3;
        var map = L.map('map', {
          maxZoom: mapMaxZoom,
          minZoom: mapMinZoom,
          crs: L.CRS.Simple
        }).setView([0, 0], mapMaxZoom);



    window.latLngToPixels = function(latlng){
    return window.map.project([latlng.lat,latlng.lng], window.map.getMaxZoom());
    };
    window.pixelsToLatLng = function(x,y){
    return window.map.unproject([x,y], window.map.getMaxZoom());
    };

        var mapBounds = new L.LatLngBounds(
            map.unproject([0, 8576], mapMaxZoom),
            map.unproject([8576, 0], mapMaxZoom));

        map.fitBounds(mapBounds);
        L.tileLayer('{z}/{x}/{y}.jpg', {
          minZoom: mapMinZoom, maxZoom: mapMaxZoom,
          bounds: mapBounds,
          noWrap: true,
          tms: false
        }).addTo(map);

        L.marker([0, 0]).addTo(map).bindPopup("Zero");

        L.marker([-128, 128]).addTo(map).bindPopup("center");

        var popup = L.popup();

        <!-- Click pop-up>
        var popup = L.popup();

        function onMapClick(e) {
            popup
            .setLatLng(e.latlng)
            .setContent("You clicked in " + e.latlng.toString ())
            .openOn(map);
        }

        map.on('click', onMapClick);

      }
    </script>
    <style>
      html, body, #map { width:100%; height:100%; margin:0; padding:0; }
    </style>
  </head>
  <body onload="init()">
    <div id="map"></div>
  </body>
</html>

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