From the Selenium website: "Selenium Core uses a unique mechanism which allows it to run on so many platforms. Written in pure JavaScript/DHTML, you copy Selenium Core tests directly into your your application webserver, allowing the tests to run in any supported browser on the client-side. Thus, you must have write access to the machine your web application server is running on to install Selenium Core."

Well, this means you can't use it to mechanize against remote/3rd party websites.

You can use remote control and need not have to use selenium core..I have not used it using https but as per the tutorials it is possible