I can speak to your first question. Per the 2018 Furman Football Record Book, the Paladins have played Army five times. They have never played the Naval Academy or the Air Force Academy. The latter makes a lot of sense because the AFA was only formed after World War II.
The first game between Army and Furman was in 1930 and the last in 1955. Each of the games were away and all were losses. I'm sure it was a nice trip by train to West Point though. Here's a brief history:
1930-Army 54, Furman 0
1932-Army 13, Furman 0
1939-Army 16, Furman 7
1953-Army 41, Furman 0
1955-Army 81, Furman 0
In the final game, Furman allowed 518 yards rushing on 56 attempts. I'm sure that a matchup in the Clay Hendrix era would be far more competitive than most of these past meetings.