There are various cases when I am away from my computer (and thus from PaperPile), and I find a PDF or a reference that I want to add to PaperPile. What I currently do is drop the PDF in a folder, make a "reminder" of the Reference in Google Inbox, then when I get back to my computer I can add the PDF or reference to PaperPile.
If there were a way to programmatically trigger an "Add Paper" option, then I could have a Folder Action either on my computer or as an AppScript in Google Drive that notices that paper or ref that I added and pushes it to paperpile automatically. I imagine that this would be addressed by the "developer API" feature request, but it could also just be a single documented POST endpoint.
Of course, maybe when the Android client is ready, I could just add the reference or PDF to PaperPile using my phone. In that case, importing from a QR code that contains the PDF url or reference data would be awesome.