It would be great to make other files available for user download associated with a video accessible through Screencast? I assume there is that possibility but for some reason I don't see how to do it. I suppose I could embed a web page on my own site that makes downloads available.
Screencast does support attachments on media. While on the view page for one of your videos you can expand the Details pane and click the Add button next to where it says "Attachments". You can see a screenshot of where this button is located: https://www.screencast.com/t/xNOlxGNx
The upload experience should be familiar if you've used the web uploader, and once you've uploaded the content and closed that window you will see they are listed in the details pane ready for download! Here's a screenshot showing where you can download them: https://www.screencast.com/t/JeeozXCKv