Settings>Notification Center> "TODAY VIEW" ...these are the only options for now, the customizations you want, I don't see a way to do them. You could disable reminders in Today View...you can still get your banners or alerts in notifications All tab. (Settings>Notification Center>Reminders)
Have you signed in to twitter on your iPhone since updating to iOS 7? Settings>Twitter> check here to see if you are signed in. If yes, have you used twitter app on your iPhone yet? If no, it may not be active.