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.
Agreed.
However, a state has the Constitutional authority to choose Electors by direct vote of the legislature. Or by a random drawing.