[SalesForce] Can users upload PDFs, Word Documents, etc. into Salesforce, possibly through the Text (Rich) field

If you are binding to a rich text field on an object, you can instead use

<apex:inputField value="{!your.FieldName}"/>

This will give you access to the Salesforce rich text editor, and allow image uploads inline with text.

I'm fairly sure that the problem here is that when you try to access the image on the email you are using the Salesforce internal link (the one that requires authentication).

When rendering PDFs you don't get this issue because the images are embedded into the file, so no authentication is required to view them.

Similar issue faced here, with rendering MS Word documents.

You could also:

1. Make the Visualforce publicly available, and give the guest user access to the data displayed (careful with this, you might want to provide some kind of authentication for your users to see the data).
2. Host the images in another service, and refer to their links in your field. Since the Rich Text Area field is just plain HTML, you can modify the image link to point to, say, Imgur or a AWS S3 bucket instance.