application can set its own badge (counter) independent of the notification scheduled. so u can surely set whatever counter you want to show the today's task by:
NSUInteger tasksCount = 10;
[UIApplication sharedApplication].applicationIconBadgeNumber = tasksCount;
talking about the notifications to show in NF bar, it is configurable by user in Device's Native: Settings-> Notifications-> so you cannot force the user to show all notifications in NF bar (lets say 50 for the day) if user has configured for only 5.
Coming back to your concern, when the app is launched, Notifications from NF bar are not automatically wiped off. you have to write the CODE to remove them from inside the app.
[[UIApplication sharedApplication] cancelAllLocalNotifications];
OR
[[UIApplication sharedApplication] cancelLocalNotification:<notification object>];