It was established in an earlier episode that Frasier's apartment is directly beneath the penthouse. In this episode we see his bedroom is two levels from the top. Either the penthouse has two stories, or someone made a mistake.
After the radio caller who watches sad movies Roz walks into Frasier's booth and picks up several blue and black cassettes, as they talk and the shot changes, it immediately changes to her holding pieces of white paper.
When Frasier described his dream to Niles in the coffee shop, he said that he saw a tattoo on his forearm that said "Chesty". However, while in REM sleep, the entire language area of the brain is less active, making reading and writing virtually impossible in dreams.
The tape machine is not connected to audio or power cables.
When Frasier is discussing his dream with Martin in the kitchen, Martin's lips are out of sync with the audio in multiple shots where he is in profile.
Frasier says "Let's shoot it to the news" but his reflection in the glass shows that his mouth is not moving.
At about 13:25, just after Daphne has finished telling the story of her dream, Niles says, "Don't stop now - this could help us". As Frasier reacts disapprovingly, Daphne gestures with both hands and her lips are clearly moving, but there is no audio. Then the camera angle changes, and she says, "Well, actually, I do have a theory of my own".