The door to Transporter Room 3 shows them on Deck 6, however, several shots of the Turbolift door down the corridor says Deck 12.
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.
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.