The 'tunnel finder' is actually a minesweeper (or metal detector), but it makes noise like a Geiger counter.
The tunnel that was found by the Gestapo was shallow enough that it would have been found much earlier. It was just a few inches underground.
After the meeting with Otto Baum at the Hofbrau, there is a cut to Stalag 13, where Baker tosses a ball to Hogan. Hogan fakes missing the ball and walks over to pick it up. The camera tracks with Hogan as he bends over to pick up the ball and looks to his left as a car drives in the gate. Hogan faked the catch (the ball was in his hand, and he just sort of dropped it behind himself) so he would have to go get the ball and see the arrival of Otto Baum - and that was the point. The camera could now track from the establishing shot of the prisoners, and follow Hogan's movement to where the ball is lying just in time to see Baum drive in.
Just after the mid-break, when Hogan, Klink, and Schultz leave to go into town (for Hogan's 'dental work'), the car drives right through the shadow of the camera. Note that this scene is cut out of the syndicated/broadcast version.
In several scenes in Klink's office, a production light can be seen reflecting off of Klink's monocle. Given the angle, the light appears to be off to the camera's right.
The reason for the Gestapo trying to take over the camp was not stated, and was not clear based on the remainder of the story.
Update: Just after the mini-credits, Hogan and the gang are discussing the Gestapo takeover. Baker asks Hogan, "What do they want with this old rat-trap anyway?" Hogan's response is, "Robbery. They cheat on camp supplies, organize a black market, move in on the local merchants - it happens wherever they take over."
Update: Just after the mini-credits, Hogan and the gang are discussing the Gestapo takeover. Baker asks Hogan, "What do they want with this old rat-trap anyway?" Hogan's response is, "Robbery. They cheat on camp supplies, organize a black market, move in on the local merchants - it happens wherever they take over."
In the opening scene, the Luftwaffe eagle and swastika are placed backwards on Sergeant Schultz's uniform. The eagle should be facing left towards the heart, but is facing away to the right. His backwards swastika is not properly displayed by showing its vertical lines on the upper left and lower right sides, but has his vertical lines on the upper right and lower left sides,