"I am flying in an airplane."
I put "w samolotem" but this was wrong. How would you say "I am flying an airplane?" (i.e. as pilot not passenger)
Should be w samolocie, but samolotem is more common.
Unless you are in an airplane, the airplane is inside a spaceship, and it's the spaceship that's flying. Lecę statkiem kosmicznym w samolocie.
Why is this Instrumental case? Do you assume there is a 'z' there and go by that?
Because Instrumental is used to express using some instument/tool and it is used without any preposition.