When Dana sits down at the kitchen table to talk with Carol, she has an empty cup and a tea bag, there's no water in the cup when the camera shows Dana reading the title of the book. Camera pans to Carol and when it pans back to Dana her cup is full of water, but never once did it show her pouring water from the teapot on the table.
When Dana sits at the table and talks withher mom she puts on hand on the book but in the next scene her hand is now holding the tea bag in her cup.
When the shower water pours down on Frank's cake, it is obvious that the cake is made of something fake, as the frosting falls off in solid pieces and the remaining cake looks something like a cardboard box.
Changing a shower head is a super easy job that anyone can do, even a moron like JT or Cody. No plumber is necessary and to actually manage to screw it up would be nearly impossible.
The coffee cup used in the airplane would not have made sense. They had just completed a barrel roll when the coffee is shown, so the cup would have been empty and it would have spilled all over everyone/everything.
When Cody rips the shower head out of the wall, the sheet rock would not have torn open on both sides of the wall, so it would not make sense that he could see into Dana's room.