iOS7 Notification center - strange "spots"

I can see what you mean, but I doubt there is anything that can be done about it.
Probably just the way iOS handles the blur effect.

 

"It's almost like I'm seeing the app icons behind the Notification Center."

That's exactly what you're seeing - it's part of the GUI.

 

It's by design. You see a somewhat blurred version of what's under the notification center (basically the screen that you were on when you brought down the curtain) when you bring it down. So on different screens it will look somewhat differently. If you don't want that you can enable the increase contrast option in accessibility settings and that will make it just plain black all the time.

 

Notification Center has some translucency which shows the app icons below albeit blurred as if looking through a filter. Completely normal part of iOS 7.

 

Try opening facetime. You see your blurry self moving around the screen