Custom template file to use themes html/page.tpl I have a custom module (using Drupal 7) that adds a page using hook_menu(). Its callback just prints the value returned from theme('mytheme_hook').

If the content of my template file is simply 'hello world' I get only a white screen with the text output.

Is it possible for this page to extend the theme's html.tpl (and even page.tpl) with 'hello world' inserted in the 'content section' of the page?

This looks like a similar thing I am trying to achieve, but it doesn't offer a solution.
If you want your content to be inserted in the content section of the page you just need to return it (as HTML, or render array) from your page callback function. Then you can override the template of any (html/page/node) level for this page by standard theme hook suggestion and customize it.