(b) No. At the federal level, the King's representative is called the Governor-General. At the state level, his representative is called a Governor.
In his constitutional capacity, the King is referred to as 'the Crown'.
Originally, in England, the Crown held and exercised all the powers of government - legislative, executive and judicial. But through a slow process of constitutional development, almost all these powers came to be exercised either by other organs of government, or by the Crown following the advice of other organs of government. For example, whereas monarchs originally decided disputes between their subjects, this function has been taken over by the courts. Similarly, in Australia, the powers of the Crown at both the federal and state levels are strictly limited.