1. Why is it spelled (not "spelt"!) AEROPLANE?
Because it uses an aerodynamic plane (as in mainplane, or wing) to generate lift.
2. Following whatever logic transpires from 1 above, why is it not spelled AEROCRAFT?
Because it is a craft which travels through the air air - craft!
3. Why is it an AERODROME but also an AIRPORT?
An aerodrome is a place where aeronautics takes place. An airport is a place where people arrive from the air - just like a seaport.
4. Who invented the damn things in the first place (Sir Percy missed his chance, unfortunately)
A moot point, personally I think that M. Bleriot deserves the most credit for coming up with the form of aeroplane that most of us now use.
5. Doesn't whoever invented a thing get to call it whatever the hell he wants and spell it that way, too?
Avion !
Incidentally, the word "airplane" is correct American english but is not equivalent to "aeroplane", it is equivalent to "aircraft". So, a helicopter is an airplane!
G