The "German" halftrack shown is actually an American M3 halftrack painted in a WW II German livery.
When LeBeau goes up on the roof to repair the leak over his bed, he is working around the exhaust stack for the heater/stove in the barracks. His bed is not near where the exhaust stack goes through the roof. When facing the door to the barracks, LeBeau's bed is to the right of the card/dining table, and the heater/stove is to the left of the table. The exhaust stack exits the roof directly above the heater/stove.
When Baker taps the bed frame to make the bed mattress go down to cover up the secret entrance to the tunnels the mattress doesn't go down it stays up. Update: When Baker taps the panel, he only taps it once (it takes two taps to activate the mechanism), and he only tapped the panel - he did not actually push the secret panel release (it did not even move) - so it should have stayed up.
LeBeau goes on the barracks roof to fix a leak because it is raining.
Later when Colonel Klink and his men are walking down a dirt road there is snow on the ground.
The rain would have washed the snow away.
The machine gun firing beside Col. Klink is not a German gun. It's an American M -1919 Browning .30 cal air cooled machine gun.
When the prisoners are playing football to distract Sgt. Schultz, palm trees are clearly seen in the background over the top of the barracks to the left of the water tower.
How did LeBeau get onto the roof? There would not have been a ladder just lying around for him to use.