With my old phone, I would be able to see any picture I took with the phone locked without unlocking it. With this phone it makes me unlock it before I can view the pictures. Is this a Moto X feature or a new Android feature (my old phone ran Jellybean)? And is there a way to disable it?
Not sure what you mean - from the lock screen I accessed the camera (using both the "twist" shortcut and the swipe left, and was able to swipe to view the camera roll just as you would usually.
I'm using the 2014 Moto X, on Lollipop.