Thank you for posting your question.
The problem is related to the WebDavModule. It is using shared verbs and protocol with http, so it's causing these problems with put/detele verbs. The resolution is simple enough though:
Open your web.config file and add the following to the system.Webserver/modules:
We are going to implement our own WebDav module, or extend the default one because of this issue.
Thanks for your feedback and links provided. We are glad that everything is fine now, and will consider these questions to be documented (in form of troubleshooting or Kb articles).