Two main questions here.

Is there a way to set that, when a new user is registered to the site, it also; 

• Sends that user's name to an external database
• Creats a MD5 hash of the password they chose as the password assigned to that database

Also, when setting up products, when an item is purchased, can we have it send that products identifier to an external database under the registered user?

Not without a custom plugin. You could shut off the internal login handler and use a database login handler instead, and then redirect new registrations to your own custom registration form alternatively.