Reminders - No Notification

My daughter and I are having problems with Reminders in iOS 10 on iPhone 6. On her phone she has recurring reminders that she had with iOS 9. These will pop up and notify on her lock screen, but the dismiss or delay options do not work - the virtual buttons are non-responsive. We added new reminders on her phone and these do not even pop up with notification.

I tried to replicate the issue on my phone - and am also not getting any reminder alerts to pop up in my notifications or on the lock screen. They sync via iCloud to my Mac and the notifications happen there - but nothing on my iPhone.

Plus - there is no "Reminders" app control in the "Notifications" setting area on my phone. (There is that option on my iPad and my daughter's phone.) Seems very strange.