1. Just to test customizing the index page I added the following line to custom_index/views/default/custom_index/content.php:

echo elgg_view('tasks/sidebar', $vars);

This worked.

2. Then I tried to create a new view for the task module. I created a new file 'hello.php' under tasks/views/default/tasks with content: (Tried also plain html)

<?php
echo '<h1>Hello, World!</h1>';
?>

I tried calling this file with

echo elgg_view('tasks/hello');

in custom_index/views/default/custom_index/content.php

But this doesn't seem to work.

The next a little bit more ambitious task is to convert the tasks widget into a view that can be called on the index page.

First of all, you should create your own custom plugin and place all the changes there. You can extend and override existing features from there. Never edit code of Elgg core of third-part plugins directly. 

Make sure to disable the caches from the admin panel while you develop new features. Most likely the view 'tasks/hello' is not working because it's not registered in the cache.

Thank you. Disabling the cache did the trick. Let's see how far I get from here :)