Stupidity: A prisoner, Jack Williams, rats out Hogan and his team to the Germans, so they fake his death and smuggle him back to London for court-martial. What a stupid thing to do. They should just shoot him and be done with it. First, what if he escaped from captivity on the way through occupied Europe? Second, he has done nothing wrong. Being a grass isn't a crime. Third, Hogan and his men are the war criminals - they often engage in combat activities out of uniform. Williams is going to be acquitted.
How to Win Friends and Influence Nazis - S3-E7
Stupidity: As Burkhalter tries to straighten the picture of the Führer, he stands right in front of it, looking directly at it. A moment later, Klink walks over and fiddles with the thing. It would be absolutely ridiculous to claim they wouldn't instantly notice the difference between a photograph of a microphone and an actual microphone inset into the picture at this distance.
Suggested correction: Not true. The crime he committed was treason, a capital offence under both civil and military law at the time. Being a partisan is not an offence under the Geneva Convention, as these were mainly captured flyers, led by a small team of specialists. To execute a prisoner, as you suggest Hogan's men should do, is murder, also a capital offence at the time.
stiiggy
Nobody from Stalag 13 accompanies Williams back to London - so where is the evidence of his "crime"? Hogan and his men often take part in combat activities both out of uniform and even wearing German uniforms, which is itself a one way ticket to a firing squad.