Have Leaflet marker popup open on page load
The leaflet module fires an event after a map is created, called "leaflet.map".
You should be able to listen for this event and get the map object like this:  

(function ($) {
  Drupal.behaviors.openMarker = {
    attach:function (context, settings) {
      $(document, context).on('leaflet.map', function(event, map){
        // console.log(map);
        // logic to open marker goes here
      })
    }
  };
})(jQuery);


Leaflet module also fires an event called 'leaflet.feature'.  

NOTE: the above code is untested