In the opening dialogue, Kevin Spacey is discussing the 12th amendment, and what happens if the states don't certify election results. He states that the House of Representatives votes to decide who becomes President, and the Senate votes to determine who becomes Vice President. He says if there is a tie in the Senate, then it comes down to a coin toss. This is incorrect. According to the Presidential Succession Act of 1947, the speaker of the House will serve as President until tie votes in both the House and the Senate are broken. It does not come down to a coin toss.
In the opening sequence, Underwood states that 73 electoral votes were needed to win the presidency in 1800. Actually, the number was 70. (Besides, if 73 were needed, there couldn't have been a 73-73 tie.)
Donald Blythe tells Claire Underwood that the Senate had chosen a Vice President due to a contingent election since 1825. However, while that was the last time that the House chose the President, John C. Calhoun won the majority of the electoral vote for Vice President and there was no contingent election in the Senate for Vice President that year. The last time the Senate chose the Vice President was in 1837 when Democrat Richard M. Johnson failed to win a majority of electoral votes for Vice President and was elected by the Senate over Whig nominee Francis Granger.
In the opening sequence, Underwood states that the election of 1800 was between Jefferson and Burr. While the House of Representatives did have to break a tie between Jefferson and Burr for the presidency, it was only because of a voting error made by one of the electors (under the pre-12th Amendment Elector College system). Burr was not really Jefferson's opponent -- incumbent President Adams was. Burr was Jefferson's running mate.
The story takes place in January 2017, when the deciduous trees in the Washington, DC area should all be bare, but there are green leaves on the trees in the Mall.
When we see Airman Webson he has only two stripes. That is almost impossible for a man of that age. More than likely he would have been an officer not a lowly airman with two stripes.