As T'Pol is leaving Trip's quarters, at the beginning of the episode, we hear the door buzz or chirp, indicating that she has pressed the button to open the door. But we can still see her hands clasped behind her back as the sound occurs.
Captain Hernandez addresses the helmsman as "Lieutenant" but her rank insignia indicates she is an Ensign.
Phlox was abducted on a city street, not on Starfleet property. this would be the jurisdiction of civilian law enforcement, not Starfleet Security. Starfleet Security would assist, but its not theirs.
At 8:32, Enterprise is seen heading north in orbit around Earth. In the background can be seen Michigan and the Great Lakes at a 90° tilt, however the map image is reversed. As depicted, Lake Michigan is on the east side of the state, in reality it is west.
The Klingons mention they could not retrieve Arik Soong from prison because it is too heavily guarded. If they are working with Starfleet, and Section 31 carries out clandestine missions, they should be able to transfer Soong to another location without raising suspicion, or at least share his paper documents from the archives. This is never explained.