How to get Login Error messages on a custom template

One way you could get the error message to your custom login page would be to use wp_login_failed hook. It could look something like this:

add_action( 'wp_login_failed', 'my_login_fail' );  // hook failed login

function my_login_fail( $username ) {
    //redirect to custom login page and append login error flag
    wp_redirect(home_url( '?page_id=4' ) . "&login_error" );  
    exit;
}

Obviously you should modify wp_redirect depending on permalinks you are using.

Then in your custom page template you can check the login_error flag and display error message:

if (isset($_GET['login_error'])){
    //print error message or do something else
}