Get node url alias/path when loading the node using node_load()
No, the URL alias is not a field nor entity property, and it doesn't get added to the node object when it's loaded.

It would be trivial to add that data though, using hook_node_load()

function MYMODULE_node_load($nodes, $types) {
  foreach ($nodes as $node) {
    $node-&gt;url_alias = url("node/$node-&gt;nid");
  } 
}