Can the path of Search module (/search) be changed without altering the module, to allow use of this path by a custom module?
If you don't like what module sets in it's hook_menu(), alter it with your own hook_menu_alter(). Example code:

mymodule_menu_alter(&amp;$items) {
  // Copy search to another place
  $items['look'] = $items['search'];

  // remove all old search settings
  // optional if you will also follow next step
  unset($items['search']);

  // add your menu item below instead of usual hook_menu
  $items['search'] = array(
    // ...
  );
}


This will unset /search path, and move all it's settings to /look path.

Worth noting is that if module generates links to /search anyway, it means it has it hardcoded somewhere. Happens. If so, you need hook_form_alter() or it's analogue to alter also these links.