For your first issue, if your trigger is “Due date is approaching” for a particular date, note that those rules only fire once just after midnight daily. If you didn’t wait for that yet, that’s why it didn’t fire. For example, if the trigger is Due date is approaching with a value of Today, then any tasks at midnight with the due date of today will trigger.
For the second issue, is there any chance you have a filter and you’re only looking at Incomplete tasks so you don’t see the completed one in Task Done?
If not, we’ll need more specific detail, like screenshots of these rule definitions and the tasks and your List view settings.