The thing with the Dropbox internal folders is that Dropbox doesn't store its starred files as exact duplicates of the original file. So a starred PDF file on Dropbox isn't stored as a PDF file (although I haven't tried renaming it to see if it still works as a PDF).
I'd like to auto-save my PDF downloads to Dropbox so that when I annotate them, I'll be able to access them frictionlessly on my computer as well.