If I the outskirts is already at maximum capacity, and I have to add 5 more monsters because of a monster surge, do I empty the Outskirts and then add 4 monsters? More specifically, does the monster that put the outskirts "over the edge" get discarded (the alternative being he gets carried over to the next group of outskirts monsters)?
Yes. See this example from the official Arkham FAQ, p.12 (emphasis mine):
In a 3-player game, the monster limit is 6 and the maximum number of monsters in the Outskirts is 5. There are 3 open gates on the board, 6 monsters in Arkham, and 4 monsters in the Outskirts. During the Mythos Phase, you draw a location that already has an open gate, so you have to draw 3 more monsters. You would first draw 2 monsters and place them in the Outskirts. The Outskirts limit would be exceeded upon placing the second monster, so you would then increase the terror level by 1 and return the six monsters in the Outskirts to the cup. Finally, you would draw the third monster and place it in the Outskirts.