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.
Stupidity: The plans Hochstetter and Burkhalter find in Hilda's desk turn out to be for the Hindenburg, a large dirigible. Burkhalter says he never saw the plane (sic) they show before. The Hindenburg was a German propaganda icon and very well known to every child in Germany. It's nigh impossible a Luftwaffe (Air Force) general would not recognize plans of a dirigible when he saw them, or not know the Hindenburg. Note that despite everything, the plans from Hilda's desk are probably meant to be not the same ones Carter later says he found inside the boxes - A Luftwaffe general not recognizing the difference between a jet plane and a dirigible is even more stupid. To avoid unnecessary corrections: Yes, according to the boxes the kits aren't for the Hindenburg but for JU-87 Stuka bombers, but that's a different mistake and not relevant for this one.
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.
The evidence is the information he gave to the Germans (actually, the spy disguised as a German major) and signed it.