I think "off", while being common, is actually redundant in; "He dropped me off at the airport". One might argue that it clarifies whether the subject was dropped, that is, released to fall, or "dropped off" which would be taken to mean let off from a ride. The more common use for "drop" is when something is dropped (released to fall) but we are happy to say, "He dropped me at the airport". In fact, the only reservation I have to not use "drop" rather than "drop off" is the association that "drop" has with item delivery and releasing to fall type meanings. I think in terms of definitions they are interchangeable and the "off" tends to sound like it's redundant. My gut feeling is that it's only there to signify a lesser degree of control, in deference to the subject being human and not an object.
He dropped me at the airport. Is fine.
After the airport drop [off] he went to the ticket counter. Is ambiguous whether you use off or not - in that we don't know if he dropped something at the airport or he was dropped. It would be better to say:
After he was dropped at the airport he went to the ticket counter or
After he is dropped at the airport he will go to the ticket counter.