In one part in the episode, after the women go shopping, Bender says, "OMG! Being chicks is so fun!" This obviously doesn't make sense, as Bender had been a female once before. It would make better sense if someone else said that. (Possibly be means that it's fun having all his/her 'meatbag' friends being chicks along with him/her.)
Petunia said, "Shut up and play four episodes of The Office!", a 21st century TV Show on NBC. However, Futurama takes place at the very end of the 30th century, and Farnsworth said in "When Aliens Attack" that all past movies and TV shows are now extinct much like the disappearance of nearly all movies from the silent black-and-white era.
Also, Fry learned to pilot the ship to impress Leela, however he might not be legally trained to.
When Leela suggests that Planet Express become an airline initially, Amy says that "...teleporters won't be invented for another fifteen years according to that guy from the future." However, in Lrrreconcilable Ndndifferences, Prof. Farnsworth is selling cheap teleporters as "disintegration rays" out of the back of comic books, and one is used to teleport both Fry and Grrl at various points in the episode.
She could mean widely available teleporters, or ones available to the public.
Scruffy keeps his/her moustache when s/he is changed to a woman
The Planet Express ship (now plane) falls out of space when it runs out of fuel. In numerous episodes previous when the ship was disabled for some reason, it would just float in space (which of course makes more sense). The ship also has propellers, which would not provide any propulsion in the vacuum of space.
Bender melts instantly in the boiling mercury. However, Bender has survived extreme heat in the past, a specific example being when he dove into the hot lava to retrieve the remains of Fry's dog Seymour. Mercury boils at 674.1 degrees Fahrenheit, lava often exceeds 1,200 degree Fahrenheit.
Bender survived the lava due to Dolomite, however his eyes did melt. Also he melted in the electric storm during "Hell is Other Robots". It's possible he just doesn't have a way to resist melting in Mercury
In the beginning of the trials to see which gender is superior the Rock Monster asks who can drink the most Arsenic, when Arsenic doesn't have a liquid state at all.