While waiting in orbit to take the first test flight, A.G.'s ship is shaking on the inside. A ship in orbit is in free fall and would not move like that.
When T'Pol states she's never read anything about 2 Starfleet pilots stealing the Warp 3 prototype, despite the history of their early warp flights being well-documented, Archer says it's not something Starfleet Command wanted to advertise. However, when they returned from the flight and were chewed out by Commodore Forrest, there were Vulcans present, who would have reported the incident to the High Command.
The fact that Vulcans were present who would, could or should have reported the incident to the High Command does not automatically mean that other Vulcans would be able to simply read about it in public, "non-classified" records; nor is it contradictory to an *outsider's* notion that Archer's career during early warp experiments is "well-documented". It's not far-fetched to assume that Starfleet Command somehow struck a deal with the Vulcan advisers to keep the incident a confidential case, out of public records.
The fact that Vulcans were present who would, could or should have reported the incident to the High Command does not automatically mean that other Vulcans would be able to simply read about it in public, "non-classified" records; nor is it contradictory to an *outsider's* notion that Archer's career during early warp experiments is "well-documented". It's not far-fetched to assume that Starfleet Command somehow struck a deal with the Vulcan advisers to keep the incident a confidential case, out of public records.
Archer receives bad news from Mount McKinley. In August 2015, Mount McKinley's name was officially changed to Denali. Knowing WHEN the Star Trek universe "diverged from our own" in irrelevant. The SHOW was written in 2002 or 2003. The writers then had no way of knowing that the mountain would be renamed more than 10 years into THEIR future.
When Archer and Robinson are preparing the NX-Beta for launch, Archer mentions that the "auxiliary APU" is not responding. APU is short for auxiliary power unit so he is essentially saying the "auxiliary auxiliary power unit" is not responding. While it is common for aircraft or other large vehicles/equipment to carry multiple APUs, they would usually be referred to as "primary" and "secondary" units to avoid any redundancy in the language.
When the NX Alpha ship reaches warp speed, stars can be seen moving past the ship but when the ship stops has only reached Jupiter and would not have past any stars as it is still within the solar system. This is one of the most common goofs in the Star Trek franchise.
During take-off of the NX Beta exhaust can be seen coming from warp nacelles instead of impulse and/or rocket exhaust port.
In the flashback sequences, Forrest has only one pip on his uniform, which should indicate that he is an Admiral, yet several times he is referred to as "Commodore".
By the time the NX-01 is ready to be commissioned he has three pips and is referred to as a Commodore. Unlike an Admiral's rank, there are no degrees of the rank of Commodore - only one.
Realistically, Forrest should have been an Admiral at the time of the NX test. Considering that there were still two more iterations of the ship to be built before the NX-01 (including the "Delta" which is the model flown by young Jonathan, with the full-size version appearing in the opening visuals of each show). This would have taken years, providing ample time for Admiral Forrest to attain the rank of Commodore.
Another incongruency is the rank of the four NX test pilots. If they had the rank of Captain during the original NX tests, years would have passed before the NX-01 was launched, giving Archer and the others a good five years during which they should have advanced. This indicates an error on the part of the writers as to military rank. It is quite common for test pilots to have an Air Force rank of Captain, but in the Navy (which is the metaphor for Star Fleet), this is the equivalent of Lieutenant. By all rights, the NX pilots should have been Lieutenants with one pip, not Captains with three pips.
The search for the first dark matter nebula would seem to be redundant as Archer mentions in Breaking the Ice (2001) that he had previously "made a run" to a dark matter nebula to set up a graviton telescope.