Corrected entry: When Bender shows everyone the card he got for Mom, he opens it and you can hear a melody play before the card says, "I love my mommy". Later, when Fry and Leela are leaving the robot museum, Leela reminds Bender to tell Mom that he loves her. Bender tells her that anyone can say that and that he has something else in store. Bender opens up the card, but this time you don't hear any melody being played. You only hear the card saying, "I love my mommy."
Corrected entry: When the Professor and Mom's clothes are thrown around the bedroom, their teeth are in jars next to the bed (they are definitely theirs as they comment on it on the DVD commentary.) The problem with this is that both of them are "wearing" their teeth, as can be seen when they talk.
Correction: Like most people, they have two (or more) pair of dentures - one is being cleaned, one is in use. Nothing unusual about that.
Corrected entry: It is revealed in a later episode that Leela wears a bracelet under he her wrist device. She isn't wearing it when the device escapes her wrist, although the bracelet is the only thing she has of her "true identity".
Correction: You have provided your own correction. 'In a later episode ...' Perhaps the significance of the bracelet (the link to her childhood) wasn't apparent until some future event (which we don't see) and she had it in her jewellery box, pocket, bank vault, whatever, until that point?
Correction: Later in the episode you find out that the card is a robot with a mind of its own so it probably didn't want to play the melody.