The API was designed with the intention that each user would link their own Dropbox account, in order to interact with their own files. It is technically possible to connect to just one account, by embedding an access token for the desired account in the app itself, like you describe, but we don't recommend doing so, for various technical and security reasons.
There isn't a way to configure a download-only permission, but I'll pass this along as a feature request. Alternatively, you could consider using shared links to link to content in your Dropbox:
https://www.dropbox.com/help/files-folders/view-only-accessYou can modify these links for direct access, e.g., to programmatically download from them:
https://www.dropbox.com/help/desktop-web/force-download
I just wanted to follow up on this to let you know that we've released "scopes" functionality on the Dropbox API, which you can use to configure an app or access token to only a limited set of functionality, such as the ability to read but not write files.
You can find more information about the release in our blog post here:
https://dropbox.tech/developers/now-available--scoped-apps-and-enhanced-permissions