You can attack a country on a different continent with your airpower alone - but if you do manage to take it down far enough to conquer it, the cost would be prohibitive.
James is correct, get rapid deployment units and create an airborne construction brigade in your home. Declare war, and send it in. 12 hours later you will have an offensive airbase in that country. Very important note make absolutely sure you are not using google chrome at this time.
To move land forces for painting the country, and doing destruction on the cheap - have military transport units standing by to airlift them when the forward air base is constructed.
As you begin attacking the different things, make sure you are using a browser supported by the game. I recently had trouble with units ignoring orders to target defenses, and opting for attacking the target instead. It got expensive real fast.
Good luck and good hunting,
the
Dont waste your GC's on a buying a C3. If your intrested in buying a country check the cash market you'll be able to pick up a much better country for around 200 GC's probably.