A newspaper published on the day Frasier was born has a story about Queen Elizabeth II giving birth. This conflicts with the chronology established for Frasier; he was supposedly born in 1952, and the queen had no children between 1950 and 1960.
When Frasier is sitting in the chair, the lid of the pill bottle is off, then in the next shot it is on then later it is back off again, all without Frasier touching the bottle.
When Frasier starts looking at the newspaper from the day he was born, he says, "Let's see what other wonders happened on that day!" Newspapers don't show the news on the day they're printed; they show the news that occurred on the previous day.
Frasier never blew out the candle on his birthday cupcake, but it is out after he sits down. Martin actually blows the candle out himself when he moves the cupcake out of the way, as he stands to help Frasier.
The "birthday" newspaper that Frasier (Kelsey Grammer) is reading is not from 1952 as there are sophisticated color pictures in it. In 1952, newspapers were printed using an expensive and involved process called letterpress and color was rare. Even when there was color, it was very simple and bright, such as in the Sunday comics. Web-offset printing, capable of more sophisticated color images, didn't become popular for major newspapers until the 1970s.
The view of the Space Needle from the living room/balcony shows just the legs of the structure, from fairly nearby. When seen in the view from Fraiser's bedroom window, in the same apartment, the Space Needle is much further away, so much so, that the observation deck is now visible as well. The rest of the Seattle skyline is also very different from the bedroom window.
Daphne gives Frasier a massage with her engagement ring on. Massage therapists normally remove all jewelry from their hands and wrists so that the person being massaged doesn't get chafed.