iPhone marking all texts as read with no notifications
I wasn't able to reply to another thread about this with the fix, so I'm making this instead in hopes that others with this problem can find it.
If your iPhone is marking all texts you receive as read without notifying you then you'll need to exit all open apps (double tap home and swipe up on each window), go into your settings -> notifications -> messages and turn off all notifications for texts. Next you need to disable iMessage in settings -> Messages. Once you've done that, restart your phone and turn your notifications and iMessage back on.
Hope this helps!