How to allow adding attachments in addition to filling in form fields and signing existing signature field

You have to allow commenting as well. Reader users will be able to add attachments as comments. You can also use JavaScript with Reader 11 and DC to automate the process a bit.

If you need Reader users to be able to add attachments like Acrobat can, you'll need to add the file attachments usage right, but you can't do this with Acrobat.