How can I create a view of child nodes based on the menu relationship?
You can do exactly this using the Menu Node API
and the Menu Node Views modules.

Create a view with your required content type, and choose a contextual filter of "Menu: Menu link ID" (this is only available with the above modules installed).

Under, "When the filter value is not available," select "Provide default value" and choose "PHP code."

The following snippet will do it, then you just expose a block from the view and you are there:

if (function_exists('menu_node_get_links')){  
   $node = node_load(intval(arg(1)));  
   if (isset($node-&gt;menu_node_links)){
      foreach($node-&gt;menu_node_links as $mlid=&gt;$data){  
         return $mlid;
      }  
   }
}