Is there any module that let (selected) users (/roles?) see the Administration pages in read-only mode? Is there a module out there which lets us assign a permission to a role which lets a user look at the site from the perspective of an Administrator, or better, from within the permissions of any selected other role; in a view-only/read-only mode? 

That would be useful to let people explore Drupal on sites with existing setup and content without having to create/clone and maintain example sites for such purposes.

(Click around in any (admin/content management) page related to the given "read-only role" without the ability to make/save changes.)
Interesting requirement.

Presumably you could assign the relevant admin permissions to the role, e.g "access administration pages", "administer site configuration" etc..

After that, you could prevent changing any settings with a hook_form_alter, adding the $form['#validate'] handler to form_set_error for non-admin users / users of that role. You could also possibly hide the submit button with a suitable #access check e.g #access =&gt; user_access('special permission'); on the button itself.

e.g

/**
 * Implements hook_form_alter
 */
function MODULE_form_alter(&amp;$form, &amp;$form_state, $form_id) {
  $form['#validate'][] = 'MODULE_readonly_admin_access';
}

/**
 * Custom validate callback
 */
function MODULE_readonly_admin_access(&amp;$form, &amp;$form_state) {
  if (path_is_admin(current_path()) &amp;&amp; !user_access('admin write access')) {
    form_set_error(NULL, t('Operating in read-only mode.'));
  }
}


It would definitely need a lot of careful testing to ensure no loopholes (e.g services API access, #limit_validation_errors cases) - make sure all bases are covered.