At the beginning when the guy calls "Spacetrak" to report the incoming object, the responding ship claims to be the only "Ticonderoga class cruiser in the area". The ship shown is clearly a battleship, not a cruiser and lacks the missile launcher indicative of a Ticonderoga class cruiser.
The microscope on the desk is definitely not advanced enough to be able to show what the two scientists are watching on the screen.
There's no way a man is allowed to enter a militarized "no entry zone" by showing just his ID, even if some months before he had some undefined and unchecked "secret stuff" clearance.
Before the press conference, Stinson says he wants to hide the deaths caused by the "meteor": considering the uniqueness of the situation that's really unrealistic to happen, especially in a big US city.
The wind speed needed to make the Space Needle fall is over 250mph making it impossible for it to be the only damaged building around.
When the scientists are looking at the head in the 'quarantine' box, you can see the gap between the glass side and the metal frame above it.
Tom discovers that breathing on the compound causes it to grow because of moisture in an exhalation. When Tom demonstrates to the others (who are all standing close and breathing) how the compound acts on water, he does so using an open container. The compound should have affected the atmosphere in the room.
There's no reason why Dimitri could not climb the stairs back and exit the hatch after opening the valve avoiding to be left there to die.