I'm using Google Drive Picker API to "attach" files in my app, but is it possible to set access permissions when a file is selected? (e.g. prompting the user to set it to public)
If you want to do the sharing within the File Picker, then I think the answer is no.
This might help though https://developers.google.com/drive/manage-sharing#launching_the_google_drive_sharing_dialog_in_your_app