Handle POST request sent from an external site for login?

1. Do I need to define a new URL for handling the $_POST request?

No, you don’t need to define new URL.

2. Which action hook do I need to use for handling such a request?

You can use init hook.

Sample code

function my_theme_send_email() {

    if ( isset( $_POST['email-submission'] ) && '1' == $_POST['email-submission'] ) {

        // Send the email...

    } // end if

} // end my_theme_send_email
add_action( 'init', 'my_theme_send_email' );

Now you can call your website with POST required POST parameters.

Update:

Executed function on init method

my code