How to make status always 'Completed' for activities added to a case?
Might be easiest to update the database is_default field. Works for me on a 4.6 site:

update civicrm_option_value set is_default=0 where option_group_id=26 and name='Scheduled';
update civicrm_option_value set is_default=1 where option_group_id=26 and name='Completed';


You may need to change the 26 but it's a stock option group. To confirm first can do:

select id from civicrm_option_group where name='activity_status';


or go to admin - system settings - option groups and see what the id in the link is. Ideally you'd be able to set default value on that screen but it doesn't seem to offer it.