I think that this is not possible in PhpED, but I will ask...

On the production server, I have a Cron job, mycron.php, where mycron.php is "below" the public area. It is not possible for a site visitor to run mycron.php, as they cannot reach it.

On my local Windows XP development computer, I cannot get phpED to run mycron.php.

The only way I can get it to work, is to create a local file, runmycron.php, which Includes mycron.php, and place it in the domain root folder

Is it possible for phpED to run scripts which are in these folders?

PhpED can't run your scripts directly even if they are in the public area. What runs your scripts is web server, or cron daemon, or shell console, anything that runs on the remote server and is able to start processes. "Public" area limits only web server's ability to run scripts. With other ways, there are no such restrictions.
With PhpED you can open an ssh or telnet console on your server and run your scripts from there. In this case it does not matter if the scripts are in "public" area or not.

The following might work for you...

Open a Command Prompt and execute

Notes: Replace C:\Program Files\Nusphere\PhpED\php5 with the actual path to your PHP installation. The quotation marks are needed if the path includes spaces.

If that works, then add it to your Scheduled Tasks in the Windows Control Panel