Are node data stored in the render array $page? Are node data (e.g. title, body, comments) stored somewhere in the renderable array $page, and converted into HTML during rendering? Or does a renderable array just contain properties that indirectly cause to load particular templates files?

Where are variables that store these values printed?
Yes, it's probably in there somewhere. However, it much easier to get the current node object (if there is any) with the menu_get_object() function:

if ($node = menu_get_object()) {
  // Do stuff with $node.
}