Friendly and Permanent URLs for documents or images don't make sense

It is because when inserting image or media - an IMG tag is used in the HTML so there can be a thumbnail displayed. This is not possible with documents so these are filtered out.
The URL format for media files of any type is being set in Settings -> Content -> Media -> Use Permanent URLs setting. ISn't it possible that the images selected are being resized? If yes, in this case the permanent URL is used automatically no matter what is set in the mentioned setting.

Yes, images get resized.

So that means there is no method to generate a permanent URL for Documents? The problem is that people move the PDFs around a lot and that breaks a ton of webpage links, so using a "permanent" URL would stop that from happening and save a lot of work.

Well, if you are resizing the images and using the permanent URLs for them, I would switch the media setting to always use permanent URLs for all media files.

Except that the Documents are not shown as media files, otherwise people would be able to use the "insert image" icon to link them, but instead they have to use the "insert link" icon instead since you cannot select anything in that library unless you use that icon. If that made any sense.

Where are the documents stored? Are they stored as page attachments or media files? As explained, when using insert image dialog, this creates IMG tag in the HTML code - which is not suitable for PDFs or other documents - that's why they are not displayed. So, when adding documents - you can link them as a standard HTML anchor tag - so you have to use the insert link dialog.

Okay, so that means that Documents can never be linked using a Permanent URL link in this CMS. Got it.

That is not true. If you set the media library setting to use permanent URLs as mentioned above, then all files linked from media libraries will be using permanent URLs. I understood that your "documents" are files in media library, right? So, if you want to use permanent URLs, set the media libraries to use them. Please, read the media library settings documentation.

If that setting is changed will it affect any webpage links that are already published that use Friendly URLs?

No, only newly added links will be respecting given setting. Already entered and saved links are not chaged.