Custom block template PHP page, possible to make content editable in UI? I have custom PHP files for a few blocks, e.g., /sites/all/themes/at-commerce/templates/block--block-10.tpl.php

Is it possible to make this content editable using the block editing UI mechanism in Drupal administration, so the customer can make changes him/herself?

For example, customer visits URL admin/structure/block/manage/block/10/configure and can edit the content.
What you're describing is already in Drupal 7 core. It's called the Contextual Links module. When enabled, modules can define menu paths that enable a set of links that is contextually relevant to the content.

In your case, a user with Use contextual links &amp; Administer block permissions will see a gear icon that produces a dropdown link from where they can configure &amp; edit content of any custom block.