Is there an API to retrieve airport departure and arrivals?
It appears that Flightaware does have an API (both SOAP and REST) that would satisfy your needs.  

Here is the documentation, see "arrived", "departed", "en route", and "scheduled".

Here's a code example from that page that gets data on flights en route to JFK and KSMO (this fits your arrivals criteria no?):



That said, you'll probably want to experiment with the API once you create an account and get a key to see if it meets all of your requirements.