The issue of the upside-down flag has been answered elsewhere, so to expand Shiz Z.'s answer on the Lincoln Memorial reference, I would say that the blood in Underwood's hand suggests that he's a ruthless, maybe violent character. He's in the place of a former United States president, so he probably wants to be the president himself.
Lincoln symbolizes (as do the statue and the Memorial themselves) important values of American politics like democracy, freedom, and civil rights. Underwood in place of Lincoln could be a comparison in positive or negative/opposing aspects. The blood in his hands and over the statue indicates the second case, so Underwood seems to be the opposite of Lincoln: autocratic and having little respect for democratic values. Also, it may be an optical illusion (caused by the color of his clothes), but Underwood seems relatively small sitting in Lincoln's chair, as if he would be a lesser person/president than Lincoln.
Besides that, replacing Lincoln in the Memorial statue has been used before to provoke surprise and reflections about the importance of some characters in the history of a nation – remarkably, in The Planet of the Apes from 2001: