How to receive email notification when an error or bad request happens?

The following is untested and unfinished, but I hope it might get you going:

In Craft CMS, there is an event "EVENT_BEFORE_HANDLE_EXCEPTION", which you could use to handle exceptions. With a plugin or a module, you could do something like:

    Event::on(
        ErrorHandler::className(),
        ErrorHandler::EVENT_BEFORE_HANDLE_EXCEPTION,
        function(ExceptionEvent $event) {
            $this->myPlugin->processException($event->exception);
        }
    );

And in your plugin services, you have that function 'processException' that should be able to read the resulting status code and act accordingly

use Craft;
use craftbaseComponent;
use yiibaseException;

class myService extends Component
{
    public function processException($exception)
    {
        $lastStatusCode = $exception->statusCode ?? null;
        
        // check for errorcodes, i.e. 400, then do something

Lastly, if you wish to have Craft send you emails, there is a built-in function for that

public function sendErrorMail($e, $myerror, $url = null)
{
    Craft::$app->mailer->send($this->getErrorMessage($e, $myerror, $url));

}

getErrorMessage just puts together the email contents and is reusable

    public $senderEmail = "[email protected]"; // or get craft website email
    public $adminEmail = "[email protected]"; // or get craft admin unser email

    public function getErrorMessage($e, $myerror, $url = null)
    {
    if (!empty($e)) {
        $message = $e->getMessage();
    } else {
        $message = null;
    }

    {# might need to set temp view path with  $this->view->setTemplatesPath(ModuleName::$instance->getBasePath());#}
    
    $body = $this->view->renderTemplate('/templates/email/error.twig',
        array("exceptionMessage" => $message, "error" => $myerror, "url" => $url));

    $senderFull = array("[email protected]" => Craft::t('module-handle','www.doe.com Website'));

    $message = new Message();
    $message->setFrom($senderFull);
    $message->setReplyTo($this->senderEmail);
    $message->setTo($this->adminEmail);
    $message->setSubject("Error on www.doe.com");
    $message->setHtmlBody($body);
    return $message;
}

I'd recommend using a service like Sentry or Bugsnag that are tailored exactly for this situation and include some Craft integrations as well:

https://plugins.craftcms.com/search?q=sentry

https://plugins.craftcms.com/search?q=bugsnag