Visible crew/equipment: In the scene where Adam walks around his apartment with his camera (because the lights were out) the filmmakers put lights in the set so the actors wouldn't walk into walls, and also letting the audience see what's actually happening. In the movie, though, there are supposed to be no lights on at all. However, just before Adam opens the closet door with Jigsaw in it, you can see the reflection of one of the set's lights on the top of Adam's camera. (01:07:40)
Visible crew/equipment: After the first call when Dr Gordon gets on the cell phone, a boom mike shadow is visible on the bathroom door. (00:10:00)
Visible crew/equipment: You can see the shadow of the camera on Dr. Gordon's sleeve when he is talking to Diana sitting on her bed. (00:34:20)
Visible crew/equipment: Just as Adam shouts "Somebody help me" right at the very start, you can see a boom shadow move over the wall. (00:01:30)
Chosen answer: Yes. You pretty much guessed correctly. We can assume that Jigsaw told Zep to tell Ali to tell Dr Gordon, "do not believe Adam's lies..." Don't forget that Jigsaw spends hours planning his tests so it would be safe to say that he had planned it all in the beginning.