How can I create public display of all Activities for Current Day + X weeks in Future (WordPress)?
You're looking at writing a custom WordPress plugin along the lines of the CiviEvent Widget plugin. Basically, you'll query the CiviCRM API to get activities (filtered appropriately to your needs) and then render them as a shortcode and/or widget.

So, yes, you can write custom code to do this, even as there is no built-in display for CiviCRM activities.