When Captain Archer goes to check on Ensign Mayweather in the zero-gravity section of the ship, Mayweather is visibly upset and at least one tear falls down off his face.
When Captain Archer goes to see Mayweather and he is in zero gravity, the captain's hair is not out of place. His hair should be at least standing up.
Travis deliberately hides the Horizon's experience with the space pirates from Archer when the latter mentioned seeing hull damage on the ship, likely to show that he thinks his old cargo ship is fine on its own. But a Starfleet officer would certainly report this right away, both so the Enterprise could be on the lookout for the enemy aliens and so Starfleet could warn other cargo ships.
When Travis enters his old room, the chess board is set up with black in lower right.