Kosinski's uniform insignia (which presumably indicates his rank as a Specialist) is fully silver when he transports to Enterprise. Once he reaches engineering it now appears to mostly black with just a narrow silver strip, similar to the 3rd black pip of Lt Commanders. However, it appears to have a curves surface, so the black and silver areas may just be the way it reflects light.
After the first jump, an outside view shows a highly dynamic astronomical scenery: gas clouds visibly moving across the screen and around a large red object (possibly that protostar Data mentions later), making about one full revolution per minute. Only massive black holes can rotate the matter around them at such speeds, but the red object looks nothing like a black hole, and besides, if the Enterprise was so close to a black hole, it would've been in very big trouble. Other than that, any astronomical view should appear frozen solid to a human observer, considering the vast distances and dimensions involved.
Kosinski never wears a communicator throughout the entire episode, even though he is in a Starfleet uniform. This is highly unusual, since the communicator pin is an integral part of the uniform; however, many details indicate that Kosinski is not a Starfleet officer: his strange single square insignia, him being referred to only as "expert" or "mister" (never by any rank), Commander Riker calling him "sir" in the transporter room, his informal (to say the least) behavior with senior officers, and so on. Apparently he is a civilian working for Starfleet, which entitles him to wear a Starfleet uniform but not an officer's pin.
At times, the illusions of a person's thoughts can easily be seen by others nearby (example: when Worf sees his pet Targ on the bridge and Tasha sees it too), but, at other times, it appears that the illusions can only be seen by the thinkers themselves (example: when Picard sees his mother in the middle of the corridor and Riker does not). Since that entire place is beyond human understanding, it's no use applying our logic to it. Apparently, the interaction between thought and reality is much more complicated than simple materialization of thoughts.
During the first jump, Data states that their velocity is "off the scale." Kosinski then says that for the next attempt he would just do what he did before; however, during each of the following two jumps, their speed never exceeds warp 1.5, according to Data's instruments. As we learn later, Kosinski wasn't really doing anything. It was The Traveler moving the ship, and he was obviously doing things very differently those other two times, because they left space as we know it. The fact that the warp at those other jumps stayed at 1.5 was mentioned specifically to let us know that The Traveller's abilities go far beyond such mundane things as "warp factors."
Kosinski wears no combadge yet his uniform is distressed on the left breast where one would be affixed. Evidently this uniform was recycled from someone who did wear a combadge and the method of application (usually Velcro) left a mark on the fabric.
Although it is stated that it would take the Enterprise-D 300 years to travel the 2.7 million light years home, the premise of Star Trek: Voyager (1995) is that it would take Voyager 70 years to travel 70,000 light years, which means the Enterprise-D is 9 times faster than Voyager at maximum warp. But, warp speed is universal and both ships are capable of high warp (greater than warp 9). It seems that, in this early episode they just didn't do the math correctly, or very probably, picked the number 300 out of thin air. There isn't a vessel capable of traveling 2.7 million light years in 300 years, except possibly a Borg cube using their trans-warp conduits.
When Picard meets his mother, the door behind him is missing half of its label.
At approximately 35:40, as the Traveler sits up in sickbay, there are several holes visible in the set behind. These are long rectangles near the ceiling, similar to air ducts, but there there are no vents and there is nothing behind them except visible supports and set lighting.
During the final warp home, the Traveler says that he needs Kosinski at the main computer in engineering, as with the first jumps. It was established that nothing Kosinski did was of any meaning; his formulas were complete nonsense and it was the Traveler's abilities alone that propelled the Enterprise on its journey.
When Kosinski enters the bridge after his 'experiment' unexpectedly sends the Enterprise to another galaxy, he begins explaining the process by which this error occurred. He states that "as the power grew, [he] applied the power 'asymptomatically'" which would imply that he applied power with no symptoms or otherwise discernible effects. It seems far more likely that the script read "applied the power 'asymptotically'", meaning that the power application tended to some arbitrary limit (the term 'asymptote' is often used in mathematics to describe the curve of a graph tending toward infinity).
According to Data, the subspace message sent to Starfleet from the M-33 Galaxy would take 51 years, 10 months, 9 weeks and 16 days to reach Earth. Unless the lengths of months, weeks or days has changed considerably by the 24th century, or the relation of each unit to one other is different in subspace, this very impressive-sounding time frame can just as easily be expressed with 52 years, 2 weeks and 3 or 4 days.
The Enterprise travels at "off the scale" velocities to the M-33 galaxy over 2 million light years away. La Forge mentions that, according to calculations on his terminal, at maximum warp it would take the Enterprise "over 300 years" to get home. According to numerous Star Trek-canon sources, Warp Factor 9 is between 729 and somewhere over 1000+times the speed of light, so it would actually take the Enterprise over 2000 years to get home, not 300. To make it home from 2 million light years away in 300 years, the Enterprise would have to travel at over 6666 times the speed of light. This is also further debunked by Star Trek: Voyager (1995), which stated that it would take the USS Voyager, a newer ship, which has a higher top warp speed of Warp 9.975 compared to Enterprise's 9.6, over 75 years to traverse over 70,000 light years at its maximum warp. Even if Voyager and Enterprise had matching top speeds, it would still take Enterprise over 2140 years to get home from 2 million light years distance, and not 300 years as stated.
Data uses verbal contractions on at least two occasions, despite the fact that he is not supposed to be able to.
When Picard leaves the bridge for engineering, he tells Data he has "the helm." The helm is the station from which a flight controller pilots the ship. When transferring command to another officer, it is more appropriate to say they have "the bridge" or "the conn," but not the helm.