It was definitely released as ‘Airplane!’ here in the UK. We can understand American English.
How about Jive?
Another Aussie - everyone I know, says and spells it aeroplane.
If I try to type airplane my spell check chucks a fit
Stuff like that (in Canada) is why I use Firefox’s British English version.
Indeed. 
To this day I have a few packets of Aeroplane Jelly in the cupboard at all times.
Why do some (all?) Americans say airplane?
Because that’s what it is. 
An airplane (or aircraft) is a plane (or craft) that flies through the air.
Kind of like how a spaceship is a ship that flies through space. You can call it a “space-o-ship” if you want, but we Americans call it a spaceship.
Now that I’ve said that, I have to go look it up on Wikipedia…
Aha! I should have known. Yet another case of the British doing things the Frenchier way.
“Aerodrome” makes me think of Mad Max for some reason.
as in “Planes, Trains and Automobiles”? 
Because during takeoff it’s airdynamic.
Sadly due to their corruption of the English language Americans are unable to enjoy an Aero bar and have to accept the pedestrian Air Delight.
But what is the proper way to say “airlines”? What are British airline carriers called? Aerolines?
No. They’re called airlines. Although “Airways” is more common in company names, as in British Airways. You wouldn’t refer to “an airway” though unless you were talking about your respiratory system.
Or more Latin/Greek (although it’s sometimes the same thing). Hydrofoil is another example.
No.
Seriously?
The American band Neutral Milk Hotel sang about an aeroplane over the sea; it scans nicely in the song, however it doesn’t appear that they were always worried about scansion, so maybe they liked the feel of it.