I think Mexicana, which started its service since 1920, is the airline in the world with longest history.

And I think the closed National Airlines of USA, which had service only in year of 2000, is the airline in the world with shortest history.

Please continue.

The oldest Airline is Amsterdam-based KLM (1919) and I believe Quantas is slightly older than Mexicana (Also started operations sometime 1920)