Yes and No to all three.
The word "Car" derives from the Latin word carrus or carrum, itself derived from the the Greek word karros. All of these terms are used to describe a wheeled vehicle. In English, the word was the basis for multiple types of wheeled vehicles, such as carts and carriages. When early modern cars were first introduce, British English used terms for the new contraption such as "Motor Carriage", "Auto Carriage", and "Horseless Carriage", though the latter two didn't catch on, it's clear that the name was derrived from the function it served (that of a carriage) coupled with the innovation in the engine of it's motion (I.E. it actually had and engine).
Of course, this being the Queen's English, we can't ignore the fact that the U.S. existed at the time nor their lovely ability to come up with new words to frustrate the English (actually many American English terms actually come from British English... the Americans just retained the original term while the Brits changed it after American adoption... though that might not be the case here). Either way, the American term does derive from carriage as well, but it seems to be that American's also invented a new word when the created the term "Automobile" which derives from the greek word "Auto" and the Latin word "Mobilius" and literally means "Self Moving". This is why Jeremy Clarkson will refer to the things related to cars as existing in "the world of motoring" while most Americans would feel more comfortable with describing this as "The world of automotives."
In all likely hood, the innovation of a personal vehicle that is similar in concept to George Jetson may start as being called a "Flying Car" or an "Aero car" early on when the concept is first introduced into the market, but might take on it's own name as the flying car replaces the car of our modern world, much as "Car" became the term accepted for the variations on comparing it to a horse drawn carriage.
And it's not unusual for terms to persist past the obscelecens of the old term. Many of the early commercially available cars were manufactured by business that built horse carriages. And the unit of power an engine outputs is measured in Horsepower. 1 HP is literally the amount of work exerted by a single draft horse.