When the construction workers come to the bar after Frasier fell in their hole, the near empty beer he had been drinking is in on the bar next to them. Half way through the scene it disappears for a couple cuts, then reappears.
Mrs. Clavin gives Woody the international sign for choking, should one of the grapes she gave him "lodge in your esophagus". Choking is caused by foreign items blocking the trachea, not the esophagus.
The actual building in Boston has no back alley, making the new rear entrance for Cheers impossible. Even overlooking this, the visible back alley implies that the pool room window is entirely or mostly above ground level, which is inconsistent with the rest of the set and exterior shots of Cheers. Building a rear entrance would require a lot more work then simply replacing the window with a door.