How do I set the HTML5 date "min" attribute?
In Drupal 8, the datetime fields are HTML5 inputs, so the popup is provided by the browser itself, or from a polyfill library for browsers that don't support these inputs (eg, Firefox).

If you want to provide a default value, you can do it from the settings for the field.  You can use the Current Date or a Relative Date as the default.  Despite the name and the docs on Relative Date, you can use this to set an actual date/time.

If you need to pull the default value from somewhere else (like config), you would hook_form_alter() the node form, and then check the field value.  If it is empty, you would set the value there.