As Klink is playing cards with Newkirk, Klink goes to put a card on the table face down. When the view shifts to show the table, Klink is moving back as if he had played the card, but there is no card face down.
In the 2nd scene, the men are trying to get a chicken to lay an egg and Hogan even orders it to lay an egg. Eventually it does lay an egg. The chicken is a rooster, so it would not lay any eggs.
When Schultz sticks his rifle into the haystack, he does not have a bayonet fixed to it. When searching haystacks, normally you would use a fixed bayonet to find someone hiding.
Klink says that he was with the 410th bomber group before transfer to STALAG 13. From a previous episode, he said that he has been a colonel for 20 years. That means that he was a colonel at the 410th, which makes him the group commander. No mention of why a long-time colonel in charge of a bomb group would get transferred to be a kommandant of a POW camp.
After Klink is relieved as Kommandant and Capt. Gruber takes over, the nameplate on the office door still reads 'Wilhelm Klink'.
There is no way that the doctor would have allowed Hogan to be present during Klink's examination.