Body and Subject fields in Mail Function
The last parameter you are passing to drupal_mail() is FALSE. That parameter is telling the function if sending the mail. The documentation describes the parameter as follows:


  $send: If TRUE, drupal_mail() will call drupal_mail_system()->mail() to deliver the message, and store the result in $message['result']. Modules implementing hook_mail_alter() may cancel sending by setting $message['send'] to FALSE.


As its default value is TRUE, just leave out that parameter, and the mail will be send, if other modules don't set $message['send'] to FALSE.