OK. The 'Group Wall' widget has a setting 'Show a form to add new posts'.

The strange thing is, it is defaulted to 'Yes', hence it should show the post widget by default but it does not (at least in my environment). You need to save it explicitly to get the posting widget displayed within the 'Group Wall' widget.

Normally hypeWall adds itself to site activity page. But on group level, it does not display on 'Group Activity' widget (at least I could not - and unlike the 'Group Wall' widget, 'Group Wall' widget does not have any 'Show a form to add new posts' setting). What would be the way to display the post box in 'Group Activity' widget?

What do you mean 'Group Wall' widget?

By default, Groups don't have widgets on its profiles but the groups' modules only.

Do you use a 3rd party plugin?

1 - Don't use elgg-data in the name of Elgg data directory.

Use data instead. (As suggestion, don't use any symbols/characters excepts letters and numbers.)

2 - Check your settings file:

/var/www/elgg/elgg-config/settings.php

Look at these params:

$CONFIG->dataroot

$CONFIG->wwwroot

3 - Try to disable system/simple caches settings on Site settings page in Administration.

Clean the caches and check you site again.

4 - Enable the logs on Site settings page in Administration:

Note: cache folder doesn't exist in data directory, caches only. The links as you mentioned are correct and not related with folders in /data directory.

More info on http://learn.elgg.org/en/stable/admin/index.html