As you found, the Dropbox API offers the ability to upload, download, and share files to/from Dropbox. It doesn't offer functionality for directly emailing files as attachments though.
If you have a file that you wish to email as an attachment, whether you happened to upload or download that file to/from Dropbox, I'm afraid that's not something we can help with. You'll need to implement the code in your app for sending email with an attachment, using third party libraries/services, as necessary.