First of all, Australia is a country, not a continent. If you are refering to Oceania, then i recommend looking at a globe or map of the world. Oceania is the islands east of Indonesia in Gunea (which is why papua New Guinea is considered part of Oceiana.) As for Caucasus issue, everything east of the ural mountains is considered Asia and not Europe. If this is true, the means the only Asian countries that are also in europe are Russia, Kazakstan, Georgia, Azerbaijan and Turkey. Dont forget, this is part of it, not all of the country.
"A continent is one of several very large landmasses on Earth. They are generally identified by convention rather than any strict criteria, with seven regions commonly regarded as continents—they are (from largest in size to smallest): Asia, Africa, North America, South America, Antarctica, Europe, and Australia."
It is possible to be both a country AND a continent. The two are not mutually exclusive.
True Sharpe, but the continent Australia is part of is commonly known as Australasia or Oceania. That's how I have read it as anyway.
If you do take the 5 continents as you put them, then where would you put Indonesia? It is hardly part of Asia, as it is not part of the same large landmass (though it is not part of Australia's landmass either). Would you make it its own continent?