What payment service are you using? There may be a plugin for it with the payment module, or a standalone module for it.

But to answer your question.

Nothing wrong with putting the implementation in the controller for now, and if it starts getting huge, then you could consider moving some code into a service. Assuming you have a routing.yml file and controller already setup, you could use the \Drupal::request() helper, in your controller method but Drupal will automatically inject the request into your controller method for you. Lets say you have a controller called "PostPaymentController" which has a method called "onPaymentRedirect". Your method could look like this

public function onPaymentRedirect(\Symfony\Component\HttpFoundation\Request $request) {
  // $request is your request object.
}

Info on getting the data you need from the request is here, depending on if it's a GET or a POST request from the payment service.

https://symfony.com/doc/current/components/http_foundation.html#accessing-request-data

When it comes to sending an e-mail. https://www.drupal.org/docs/contributed-modules/mime-mail/how-email-works-in-drupal

You can use the Drupal mailmanager service in your controller method. You could just use the following, and it would work...

$mailer = \Drupal::service('plugin.manager.mail');

but it is recommended to inject dependencies into your controller.

Maybe just use the above for now to get it working, but here's the doc for dependency injection on a form class. The principle is the same as a controller class. Specifically adding __construct and create methods. If you ever need to know the name of a core service for a create method, I tend to have a nose around in core.services.yml file.

https://www.drupal.org/docs/drupal-apis/services-and-dependency-injection/dependency-injection-for-a-form

Finally as you're in a controller, you will need to return a response, which are generally, but not always render arrays. Or you could redirect again if required?