Chekhov's Skill: See above: Whitmore's tour of duty as a fighter pilot in the Gulf War.
The Eternal Churchill: Right before the final battle, President Whitmore gives one of the most memorable speeches in movie history, a speech that seems to be deliberately designed to be similar to Churchill's refusal to surrender.
Everyone Calls Him Barkeep: OK, show of hands, who actually remembered that the character's name was not "Mr. President" without having to look it up?
A Father to His Men: Becomes indisputably true during his brilliant Churchillian speech right before the final battle.
Number Two: To President Whitmore. At the beginning of the movie, he requests to stay at the White House alongside the President (it turns out to be lucky. The rest of the JCS were apparently killed by the aliens after they evacuated to NORAD). Throughout the movie, Grey is unfailingly loyal to Whitmore and might be either the new Secretary of Defense or even the new Vice President at the end, given the fact that the former VP is dead and the former Secretary of Defense was unceremoniously fired.
Obstructive Bureaucrat: Does nothing but provide bad advice to the President, and withholds crucial information (regarding the Roswell crash and Area 51) that could have been more useful had it been divulged earlier.
The Scrappy: Both to viewers, and in-universe, to the point where a furious and exasperated President Whitmore finally fires him.
Skewed Priorities / The Sociopath: It's outright stated in the book that he cares more about his political career and legacy than saving the human race from extermination.
What an Idiot: You're the Secretary of Defense and an entire fleet of alien spaceships shows up and float ominously over the world's major cities. As the former head of the CIA (it's stated somewhere in the movie that Nimzicki was this), you became privy to the fact that Area51 has indeed been housing a crashed flying saucer and alien corpses and been studying them and their tech for decades. Do you choose to immediately inform the President about everything the research had discovered in order to better inform his decision-making? Apparently not.
Debatable; while he makes (in retrospect) bad choices, his working assumption that launching nuclear weapons will defeat the invaders remain untested for long, and it isn't exactly unreasonable (at least before the first counterstrike with fighter planes fails) that it could work. If nuclear wepons would've worked, there would be no need to reveal Area51.