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, two weeks and 3 or 4 days.
The Enterprise is transported at "off the scale" velocities to the M-33 galaxy over 2 million light years away. Geordi La Forge mentions that according to his 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 ranges between 729 to 1000+ times the speed of light, so in actuality it would take the Enterprise over 2000 years to get home, not 300. In order for the Enterprise to make it home from 2 million light years away in 300 years, it would have to travel at over 6666 times the speed of light. This is also further debunked by Star Trek: Voyager, which stated that it would take the USS Voyager (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 Geordi stated.
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).
Kosinski's uniform insignia (whatever it indicates) is filly silver when he transports to Enterprise. Once he reaches engineering it is now mostly black with just a narrow silver strip, similar to the 3rd black pip of Lt Commanders.
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 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 wearing 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, Tasha sees it as well), but at other times, it appears that the illusions can only be seen by the thinkers themselves, and others cannot see what they see (example: when Picard sees his mother in the middle of the corridor, Riker does not see her). However, 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 Traveller's abilities go far beyond such mundane things as "warp factors".