The rules state that as soon as a player's network grows to include the other city listed on the Airmail Route card, he discards the card and takes $40.
The 'Place a Plane' section suggests that a player's 'network' is their planes on the map.
I'm assuming from this that to claim an Airmail Route card, the starting city and destination city do not have to be linked through adjacent cities with that player's planes. For example, could I use a medium or large plane to place a plane in the destination city and claim my Airmail card, even if the starting city and destination city are not directly linked through adjacent planes?
That's correct. As soon as you have an airplane in both the starting and destination cities, you collect the $40.
It's also important for new players to remember that players do not have to satisfy their Airmail Route card in order to win the game. It's strictly a $40 bonus for making the connection.