For the first, you may want to implement a reminder (setup->Email Notifications->Reminders) for late tasks. This way people will get a notification x days after the task was due.
That said, this is a change management problem. Sounds like your group needs to better understand WHY the tasks need to reflect reality. Create awareness of the need and the "What's in it for me" for those doing the work. What value does the individual get from it, does the organziation get from it. You may also want to send automated reports out to the department managers of overdue tasks --- if the people arent concerned, create the deside to get the managers off their backs.
