The user decides
To answer your last question first:
I tried adding the plugin inside the theme plugin folder [...] What am I doing wrong?
You are trying to bundle functionality into a theme. That is the wrong thing in the first place. Themes offer presentation - Plugins offers functionality.
Aside from that, functionality should be the users choice. And your theme should work with and without the plugin. You also save the user from loosing his settings when he switches the theme. Just note that your theme "supports subheading plugin". The installation and activation is up to the user.
To achieve that, you can just wrap plugin functionality in a filter or hook and check if the plugin is active using
add_action( 'subheading', 'subheading_callback' );
if ( ! is_plugin_active( 'plugin-folder-name/plugin-file.php' ) )
remove_action( current_action(), __FUNCTION__ );
// Call subheading-plugin function to output subheading here
// From the subheading plugin:
// $before = '', $after = '', $display = true, $id = false
the_subheading( '<h3>', '</h3>', true, get_the_ID() );
Then in your template file, just add the hook to output the subheading.
// for e.g. single.php - in the loop
do_action( 'subheading' );
Aside from that, there is the "TGM Plugin Activation" script which you can use in your theme. It allows you to tell the user upon activation that s/he needs the plugins X, Y and Z which you can set up with default settings (IIRC).