Continuity mistake: When Marco is in the library searching the internet for information about the scientist, he isn't wearing headphones. Then he notices the security camera recording him, and in the next shot, from the camera's POV, he's wearing them. When it cuts back to 'our' regular view they're gone again immediately, and he puts them on to listen to the computer.
Continuity mistake: In the scene when Liev Schrieber is with his mother before they go out to the party, she kisses him, than starts to lean forward to kiss him again. But when the camera cuts to the reverse angle, she hasn't moved an inch. It's as if she lunges back to her original position.
Answer: According to the Constitution, the Vice-President elect is actually elected separately by the Electoral College, i.e. they are really not a package deal, the death of one doesn't negate the election of the other. The Vice-President elect would indeed become the President.
Myridon