The door to Transporter Room 3 shows them on Deck 6, however, several shots of the Turbolift door down the corridor says Deck 12.
In the third season, the main characters' uniforms are redesigned, eliminating the shoulder piping. Early in the season, the uniforms also have two noticeable seams down the center of the front, but a further redesign removes those seams in later episodes. In the penultimate scene of the episode, Picard wears the version with the visible seams. When the room returns to normal, his uniform now lacks those seams.
While Jeremy is looking around his apparent home on Earth, he calls the cat "he". The cat is a calico (three-colored). Almost all three-colored cats are female. A male calico would usually have to have a moderately rare (1 in 500-1000) condition where they have three sex chromosomes (two X and one Y chromosome) in order to be calico. However, a third of male calicos do not have this chromosome abnormality.
There is no way to know the true sex of the cat used.
Worf explains to Picard that Lt. Aster was walking 3 meters behind him and an explosion went off with no warning, that she took the brunt of. The damage to his uniform, and the wound on his chest, would indicate the explosion would have had to have happened in front of him, something not possible if it was several meters behind and no warning was given.
After Riker is given the bridge, he is later seen sitting in his standard first officer's chair, rather than the captain's seat, where whomever is in command (even temporarily) would sit. This seems to have been done to get Riker, Data, and the acting tactical officer in the shot together.
Jeremy is in his quarters watching a video of his mum. In the video, as she comes round the corner, she says to Jeremy "I'm getting closer" but her mouth does not move.
In the teaser, Troi begins to sense the danger to the away team almost half a minute before it happens--we see two reaction shots of her *before* the explosions start to happen. However, as she is neither a precognitive (which would allow her to see the away team's future) nor a clairvoyant (which would allow her to see the trap awaiting the crew), there should be nothing for her to sense before the away team is attacked.
Jeremy Aster was allowed to stay in his quarters despite being an 11-12 year old child whose only guardian had died. Surely an advanced culture like the Federation would not leave a grieving child to be unsupervised or unsupported after the death of their only remaining parent.