Revealing mistake: In the mess, you can see everyone has their food on metal trays and ceramic dishes. Sgt Zale, though, has his lemon pie in a polystyrene dish - so that it doesn't hurt Klinger when he gets it in the face. (00:08:15)
Audio problem: During the fight in Rosie's bar, a soldier is punched, and then thrown through the opaque window with the wood frame behind him. There is a sound of breaking glass accompanying the motion, but you can see the torn paper.
Answer: Even an Army field hospital would have SOME weapons for self-defense if the enemy attacked. What's meant is that it's not a combat outfit, and therefore wouldn't have enough weapons to actually attack anyone. (It's also mentioned a few times that the Officer of the Watch is supposed to carry a pistol at all times, but Hawkeye refuses to do it when he's in charge and Colonels Blake and Potter don't enforce the rule).
Captain Defenestrator