Deanna Troi is able to sense deception and evil intent on the part of DaiMon Bok. Earlier in The Last Outpost (1987) Troi states that she is unable to sense the Ferengi's emotions, she says that they may be able to shield their minds from telepaths. It is later established in Trek lore that Ferengi are immune to Betazoid mind-reading and empathy. In Ménage à Troi (1990) Lwaxana Troi, Deanna's mother, says it is thought that Betazoids can't read Ferengi's thoughts and emotions because of their four-lobed brain, she also states in The Forsaken (1993) that Betazoid telepathy doesn't work on any species with a four-lobed brain.
Just before the Ferengi beam over, the crew is shown standing at parade rest, hands clasped behind their backs. The next shot shows them from behind, with their hands at their sides.
After the Enterprise discovers the drifting USS Stargazer, there's a space shot of the Enterprise with the Stargazer just a few hundred meters below it, but, in the next space shot, the Stargazer is shown by itself with the Enterprise nowhere near it.
When Picard is speaking to the Enterprise crew from the bridge of the Stargazer, he is seated in the captain's chair and pulls the armrests down over his lap (the closest thing starships of that era had to buckling a seatbelt). When the transmission cuts off, he is instantly standing and walking around the bridge.
The Enterprise locks a tractor beam onto the Stargazer while its shields are up. This procedure is not uncommon though. What usually disrupts a tractor beam is a remodulation of the shields, not the shields themselves.
When the Enterprise is preparing for an attack from the Stargazer, Kazago appears on the main viewer unexpectedly and holds a brief, unwanted conversation with Riker. An outside transmission should not be able to just appear on the viewer like that without an Enterprise crew member accepting the incoming communication and displaying it, which would not be done without the commanding officer's authorization.
Obvious stunt double for Patrick Stewart. When Picard destroys DaiMon Bok's mind control device, the double goes backward from the blast (side of his face and above the neck area; obscured from the camera angle).
Picard looks into a trunk that contains some of his possessions, as if they had been packed. Picard had been forced to abandon the Stargazer so quickly it was very unlikely he had had time to pack anything. If he HAD had time, he would have taken those things with him.
The Picard Maneuver is meant to utilize the element of surprise by tricking an opposing ship's sensors into seeing two of the same ship, getting them to fire on the first (phantom) ship while the second (real) ship fires at point blank range. After the Stargazer performs the Picard Maneuver, Picard is still seen on the bridge telling his (imagined) weapons officer to hold and wait, well after the phantom sensor image his disappeared, thus negating the entire point of the maneuver.
When confronting the Ferengi first officer, Kazago, Riker tells him that, "You know who controls those spheres," plural. At that point, the Enterprise crew was only aware of the existence of one of the spheres.
With Picard suffering from such a bad headache (which have become almost unheard of), Dr. Crusher would surely have taken him to Sickbay to rest where she could keep an eye on his condition and run more tests to find the cause.
After hearing the doctored log, Riker and Picard say that it will take a full day for their subspace communication to reach Starfleet, and another full day for Starfleet's response to reach the Enterprise. In all other instances of subspace communication between the Enterprise and the fleet, communication is instantaneous, enough so that casual conversations can be carried out.
At 8:26, while Picard introduces his officers to the Ferengi on the bridge, a boom microphone lowers into view for two seconds.
When Wesley Crusher rushes onto the bridge, he informs Commander Riker that they should be receiving an intruder alert, which is untrue, since this means someone boarded the ship without permission. The correct term would be a perimeter alert, which is an unknown ship within the vicinity.
Picard introduces Data as his second in command. Riker is second in command while Data is third. He probably meant Data is the second officer.
Picard has been having headaches and states that he smells something burning while in sickbay, which he dismisses as a memory from his last battle aboard the Stargazer. Smelling a burning odor is a common sign of a stroke. That, combined with the unexplained headaches, should have caused much greater alarm for Dr. Crusher, yet she simply allows Picard to resume his duties.
In his quarters, Dr. Crusher injects Picard with a sedative without first informing him or gaining his consent. It is only after she has injected him and he specifically asks what she injected him with that she explains it. This is unethical for a medical professional.