How to display a Flag in a tpl file for different View modes?
You actually can display a Flag link as a pseudofield, which can be ordered among other entity elements in the Manage display settings for an entity type. Go to admin/structure/flags and pick the Flag that you need. Next, check Display link as field in the section Display in entity links. You can now use the link in your template files by rendering the node and getting the right field (use Devel for this).

Alternatively, you can output a Flag link by using flag_create_link($flag_name, $entity_id). In that case you don't need a pseudofield.