The first scene is set on August 12, 2152 and the menu on the galley computer indicates that it is Thursday. August 12, 2152 will be a Saturday. (While it's conceivable that calendar rules could be rewritten in the future, it is highly unlikely.)
In her analysis of the harmful unknown radiation, T'Pol states "The radiation stretches outward half a light year in every direction except one." By definition, all radiation in a vacuum "goes on" forever though until it meets an obstacle that blocks it. However, its intensity diminishes. It gets "diluted" as distance from the source increases, a result of elementary geometry described by the inverse-square law. The interesting question is how far you need to stay away from the source in order to keep the intensity to tolerable levels. Even more specifically, you look at radiation doses, and to keep those down, you need to include the time variable. A course "between the stars" is not going to be much help. What they'd actually need is either a course that keeps the total radiation dose minimal and that would most likely include making use of the object that blocks the radiation in the one direction where it "doesn't extend".
While dictating the preface of his father's biography, Archer tells the computer to "delete the whole thing." The computer immediately beeps, indicating the order was carried out. No computer programmer would make deletion so easy. The computer would almost certainly ask something like "Are you sure?" and give at least one fail safe option.
However, it's becoming more and more common to have a 'delete immediately' function with a 'recently deleted' area for simple data recovery, as continuously having to affirm each deletion two or more times is exceedingly annoying.
Trip is using a two-step regulator off the air tank he is using to power his pneumatic tools. The second needle is indicating zero pressure, yet the tool is working fine.
In the text Archer is writing about his father, there's a typo near the bottom: "his won" should be "his own."
When the doctor is about to perform brain surgery, he says he's going into the pre-frontal cortex to shave some of the occipital lobe. Those two parts of the brain are in separate locations. (It should be remembered that his reasoning facilities are not in top form, allowing some understanding for such an error.)