Seven gets angry over the wager The Doctor and Paris make. Notice the first camera shot when she leaves the mess hall. She walks in the direction of Neelix's back. The next shot looks like Seven needs a much longer path to exit the mess hall. Somehow Neelix is in the right of Seven.
When Seven of Nine sings the vocal exercise with the Doctor, the display screen shows a "simple melody." The letters say that the melody consists of the notes "A C E C D B A G A." This does not correspond to the music notation displayed on the screen, which shows the melody containing C sharps and G sharps. Seven sung it as such.
Even in the 24th century, intergalactic travel is beyond the abilities of the Federation, so an image looking back on the "Milky Way" would be impossible. During The Doctor's talk with Seven about sexuality, he shows an image he calls "The Milky Way."
Tomin gets drunk on Synthohol - a substance that, in Star Trek canon - tastes like alcohol but does not have the intoxicating effects; however, the Doctor says Tomin's physiology does not have the necessary enzymes to break down Synthohol. This leads to a hangover and could lead to intoxication as a result.
When The Doctor prepares to extract nanoprobes from Seven of Nine, he refers to Thomas Paris as "Ensign Paris." This is factually correct, as Paris was demoted to ensign in Thirty Days (1998) and was not restored to the rank of lieutenant until the start of Unimatrix Zero (2000), at the end of Season 6.
At the ambassador's reception, Ensign Paris is seen in dress uniform but the Doctor is not.
As the Doctor is a member of the senior staff (despite being a hologram, he is treated by the crew as a sentient individual and a department head) and Paris is a disgraced ensign and helmsman (a position not typically considered a member of the senior staff, despite his status as a member of the show's main cast), it would be more appropriate for the Doctor to wear a dress uniform than for Paris to wear one.
As the Doctor is a member of the senior staff (despite being a hologram, he is treated by the crew as a sentient individual and a department head) and Paris is a disgraced ensign and helmsman (a position not typically considered a member of the senior staff, despite his status as a member of the show's main cast), it would be more appropriate for the Doctor to wear a dress uniform than for Paris to wear one.
In the holodeck, when the ambassador is very drunk, Neelix mispronounces synthehol and says synthedol.