Is there a module or method for processing and generating codes for music download cards?
Well I'm unaware of a turn-key (Drupal Distribution) for a Dropcard ecommerce site per se. The components exist to make such a site using some custom code. Some example modules that fit your use-case are:

TokenAuth


  Token Authentication provides a token for use in the URL to
  authenticate users to your site. It only works on paths configured in
  the settings page.


Commerce File


Ability to limit access to the file with limit settings
Default limits: duration, download count, IP address count
Limit default values can be inherited at the product level such that the product always uses the current default value
After purchase of a file, limits are statically recorded to the line item so that they do not reflect any changes to the product's file limits.
A user's limits reflect changes to an order, even after order is completed.
Order deleted, canceled
Line item deleted, update
Utilizes Rules to control workflow for:


File download access: controlling access and reacting to access denied
Order workflow: issuing access licenses and revoking access licenses in the order



Views Data Export -- export your authentication links to paper cutouts.


  This module is designed to provide a way to export large amounts of
  data from views. It provides a display plugin that can rendered
  progressively in a batch. Style plugins are included that support
  exporting in the following types:
  
  CSV, Microsoft XLS, Microsoft DOC, Basic TXT, XML.


Commerce Module


  uhh make an eCommerce site using drupal.


Many of these modules likely make use of Drupal core concepts for Entities, Cron Jobs and likely Queues. You may find the Queue API useful to build the authentication tokens for example every hour for new audio content.

The ingredients exist for a developer to create what you're asking for, good luck.