I haven't read the case, but where does the case go after it becomes a 2-2 decision?

Not quite. I would think that relatively few judges would owe quite so much to a single contributor, and of those, I would hope the vast majority would have the decency to recuse, or (as in Mississippi) would be forced to recuse by their fellow judges.

the highest decisions of the state courts are reviewable by the SCOTUS.

Only on issues of federal law. The merits of the Caperton case were, IIRC, purely state law. The only federal issue was the due-process issue re: recusal, which as we can see was indeed appealed.

There is a better approach, the highest decisions of the state courts are reviewable by the SCOTUS. If a party aggrieved by the decision believed that due process was violated because the decision was in error due to financial bias on the court, the SCOTUS can review it to see if there was corrupt influence or if the decision was supported by the record.

The problem with recusals is that it works against one of the main protections against bias on appeals: a five-to-nine member panel.

I think of the course the problem is that there isn't a clear line to judge when recusal is necessary. Personally I am not a huge fan of elected judges, but there are problems that come with an appointment process as well-I suspect appointment processes are more expensive and elections are cheaper, so at a local level elections may make more sense.

There was a study done pretty recently that showed pretty conclusively that elected judges side with in state parties over out of state parties a highly disproportionate number of cases. I don't remember if it compared the number to the rate of appointed judges or not.

I agree that the larger reasoning in the opinion begs a lot of questions, but the decision itself seems pretty obvious.

But how does one know what a higher court will judge “exceptional” a priori?

Classic Kennedy, the only one who knows the line is him. He is the decider.

“Just as no man is allowed to be a judge in his own cause, similar fears of bias can arise when — without the consent of the other parties — a man chooses the judge in his own cause,” Justice Anthony Kennedy said for the court.

...

Doesn’t this logically mean you can’t have elected judges?

I wonder; Might that principle also be extended to include a judge who makes statements which could arguably be taken as pushing the agenda of group identiies of whch she claims membership?

Such as, Female and Racial Minority, for example?

Hmmmm.