At the President's farm house, in several scenes, the US flag is on the right side of the entrance and the Presidential flag is on the left. The US flag is always on the left.
When Donna (Janel Moloney ) enters Josh's room to find him not prepared for the presidents upcoming speech to kick off his reelection campaign, Josh (Bradley Whitford ) is wearing a Wesleyan University shirt. The character Josh did not go to Wesleyan, he went to Harvard for college and Yale for law school. Though Bradley Whitford went to Wesleyan, the character Josh may have acquired the shirt in other ways.