When Detective Larsen is watching the video sessions of Ray and his doctor, the video appears to have been shot by a camera placed on the doctor's desk. Shots of the footage alternate with shots of Detective Larsen watching it. In the first shot of the footage, the camera is aimed at Ray and the legs of a guard are visible on the right side of the screen. A few seconds later, footage from the same interview is much more zoomed out, showing the whole guard.
While a foot search may have been undertaken, a helicopter was present. Since most police helicopters have Forward Looking Infra Red (FLIR), the helo would have been used to look for bodies in the woods and the ground personnel sent to them.
When Detective Larsen drops his gun in the cabin, he drops it onto a carpet. However, the sound is like a gun dropping onto hardwood.
Beth has been shown to be a kung fu master fighter/Mary Sue, easily able to take down larger male opponents. Additionally, she is a trained LAPD officer. After taking Ray down, she would either have ganked him using her feet such via something like a head stomp or after freeing her hands she would have handcuffed him. Instead, she acts like a frightened little girl and tries to run away. Next, when she is alone on the bed instead of pulling her hands in front of her over her feet and then freeing her ankles, she just lays there. The first error is repeated later to the point of ridiculousness.
LAPD policy generally prohibits officers from surrendering their weapons. Jack as a veteran officer would know this. Since help was on the way, an armed standoff was in his favor.