President Bartlet exclaims that people in Maine have always been iffy about him and he doesn't know why. Martin Sheen played the President in The Dead Zone, in which he is fated to start World War III using nuclear weapons, bringing about the end of civilization. The Dead Zone was written by Stephen King, who hails from Maine and sets many of his stories in Maine.
At the end of the episode, the President asks CJ to move the press "off church grounds" so he can take questions. The National Shrine of the Immaculate Conception is an enormous structure at the center of a sprawling campus, and is also in the middle of Catholic University of America. The nearest spot "off church grounds" is a couple of hundred yards away and on the other side of Michigan Avenue. Under normal circumstances the Presidential motorcade would have to take him that distance.
The football game being played on the TV while the President (Martin Sheen), C.J. (Allison Janney) and Sam (Rob Lowe) talk about the debate format is a Canadian Football League game between the Calgary Stampeders and the Edmonton Eskimos (now called the Elks). Both teams are from the province of Alberta and have been rivals for over sixty years. They also are consistent challengers for the Grey Cup (CFL championship). At the time of the airing of this episode the Stampeders had just won the Grey Cup and the Eskimos had achieved their 30th consecutive playoff appearance. The quarterback (#12) for the Edmonton Eskimos is Jason Maas, which makes the game a contemporary of the time the episode was filmed as he played for Edmonton between 2000 and 2005.
There is no such thing as the "Medal of David." The closest equivalent would be the Jabotinsky Medal, or the President's Medal (first awarded 2012).