Hello,
You should actually be able to check downloaded-files’ events from WP-Admin > Private Area > Logs.
Also, the Notifications add-on can send emails about such events to site administrators, author, or recipients.
We don’t have a way out-of-the-box to display download counts. Best way would be to hook into our functions and edit our templates (basically, add a post_meta to store the file’s download count, and display that on your button). This can vary depending on what you really want to do.
Best regards.