[sourcery]

Who decides whether or not a state’s legislature has actually appointed any Electors? And who decides who—if anyone—has actually been so appointed?

Constitutionally, it would appear that Congress has the final say on that. Congress is Constitutionally responsible for counting the votes of the Electors.

[jjotto]

Congress gets the final say. See the election of 1876.

[FredZarguna]

1) The state cannot choose its electors using any process that violates the US Constitution (and necessarily any of its Amendments.)

So, for example, if the statute says that electors will be appointed by a vote of all black transsexuals over the age of 61, the Supreme Court would strike down their selection as a violation of the Equal Protection Clause. (This is the basis on which they actually struck down Gore's proposed method of counting only specific kinds of votes in specific counties in Bush v. Gore.)

2) If the statute passes US Constitutional muster, only the Congress would be able to deny the electors' appointment. As a practical matter, that isn't going to happen.