When Commander Ransom enters the conference room, Tendi is holding a lint roller. She puts her hand behind her back and the lint roller disappears. It is never seen again, and there is nowhere on her person to conceal it.
The stolen Romulan bird of prey is visible in space. After a brief interior shot, in which it is announced that a Romulan Warbird is scanning, the exterior shot shows the bird of prey cloaked. No one actually cloaked it and the amount of time between when they detected the warbird and the exterior shot would not have given them time to engage the cloak.
Just before the lights in the silo come up to reveal it was a party, silhouettes of decorations can be seen in the darkness along with the silhouettes of the trial observers. The silhouettes of the observers could be plainly seen throughout the lower deckers' testimony yet there were no balloons.
Captain Freeman tells Ensign Mariner to send the alien ship "a message," which Mariner misinterprets as firing phasers when Freeman meant to send them a literal invitation to dinner. Mariner is stationed at the ops console. Communications and weapons are both controlled by tactical, where Lt. Cmdr Shaxs is stationed. Freeman should not have asked Mariner to send the message nor should Mariner have been able to carry it out, whatever the meaning.
Rutherford is said to have messed with the red alert sirens in the shuttle bay, which resulted in the four ensigns not having been aware that the ship was at red alert. The lighting should still have turned red, notifying them of the alert status.
While blacked out, Rutherford disables a squad of Vulcans by performing a nerve pinch on them. The Vulcan nerve pinch is performed by harnessing neural energy and directing it into a target, something only Vulcans are physiologically capable of.
When Boimler and Mariner discover that the ship is at red alert, they head to their stations on the bridge. The bridge would have been fully staffed when the red alert was called. Personnel do not swap out when a ship is at alert status.
When Rutherford is standing on the Romulan Bird of Prey, he is far too large in comparison to the ship's size. A bird of prey has a wingspan of almost 200 meters yet is size compared to Rutherford would make it roughly a tenth that size.
Ransom uses the map of the Neutral Zone to plan a special operation. As Mariner pointed out, Starfleet would have no need of such a map, as they have advanced stellar cartography systems. Additionally, the map shows the location of Romulan ships. A static map would not be able to accurately display the current positions of movable objects such as ships.
The stadate of when Freeman obtains the map is 57818.4. The star date of when Rutherford, Shaxs, and Billups retrieve the Romulus Bird of Prey from the museum is 57791.1. The former event should've occurred before the latter, not nearly one month after.
Ensign Tendi is roped into a classified mission after being mistaken for a special operative while cleaning the conference room. Tendi is in medical division. Cleaning the conference room would fall under ops or perhaps even bridge crew, never medical.
Tendi is mistaken for a special operative known as "the Cleaner" because she is in the conference room for the planning of Ransom's special op. Ransom never makes the connection that she is not the Cleaner, in part because the real Cleaner never arrives. The fate of the actual Cleaner is never revealed. If a Starfleet special operative who was meant to be part of a black ops mission went missing, Starfleet should have investigated their disappearance and informed that operation's commander.
Ransom has had direct contact with Ensign Tendi, so he should be able to recognize her as a member of his crew, not a special operative. At the very least, the fact that she is in full Starfleet uniform when the other members of his black ops team are in plain clothes should have clued him in that she was a regular member of the Cerritos' crew, not a special operative.
Q refers to the Cerritos' senior staff as "mes amis" (French for "my friends"). Q only sprinkled his dialogue with French words and phrases on TNG because Captain Picard was French. There would be no reason for him to speak French to the Cerritos' senior staff, as none of them are French.