If you had Photoshop, you might (possibly) be able to create a watched folder/droplet that automatically opened up a PDF that you export from Google Docs, rendered it, flattened it, trimmed the white space, exported it as a web-optimised JPEG/PNG and saved it in the appropriate Dropbox folder with the appropriate filename...
If you had Photoshop, you might (possibly) be able to create a watched folder/droplet that automatically opened up a PDF that you export from Google Docs, rendered it, flattened it, trimmed the white space, exported it as a web-optimised JPEG/PNG and saved it in the appropriate Dropbox folder with the appropriate filename...