Send notifications even when an App is "open". I don't know if this is the case with all apps, but a lot of the time I won't get a notification for new messages if I've left that App "open" on the screen and locked the device. I have to switch to another app, or go to the home screen before locking to get them.
I agree with most of your points, but this one specifically is a problem with how the app(s) are developed, not with iOS. Most apps ignore notifications if they are open. If your app is open you can trap your own notifications and do something with them but most apps don't bother since they have some other mechanism to show you what the notification would have said.
Alternately the developer could have decided that when you lock the device to sleep the app so that iOS would handle the notifications in the normal way. This isn't really something that Apple should fix, it's something that apps should design in already.