Mail, messages, only show up in the lock screen (as single alerts) if they haven't been read. Reminders don't remind (or come up in the lock screen) until the time you set comes to pass. Once you unlock the device, they disappear from there.
The 'View in Lock Screen' option for everything is only for new items which come (emails, reminders, messages) in when the iDevice is locked (or asleep).
The banners option means that a small banner comes up at the top of the screen, then goes away after 5sec. This only happens when the device is unlocked, not in the lock screen (cos the device is locked and u won't c it).
If you want to see all reminders, unread messages (doesn't seem to work for emails you mark as unread) and events, swipe down from the clock, (menu bar at the top of the ipod) to see all your notifications. (This will not work on the lock screen).
Hope this helps. 😝