"What is the significance of this to allowing Sharia Law to determine contract compliance?"
Because it's an example of how "choice of law" applies in a religious-law context. This, ironically, is the link that is most applicable to the case at bar (or at thread). The fact that the court of arbitration is applying religious-based law (as opposed to the law of a foreign sovereign, like Germany) is irrelevant. What matters is American jurisprudence recognizes the legitimacy of binding arbitration, and allows choice of law to be applied in that arbitration.
"Regarding the fourth link, I suppose the intent was to read the 196 reader comments and not the advert for the radio show..."
Yes, sorry. For that particular site, some of the most valuable information and contribution comes from the posters themselves. This is actually by design of the forum management. Many of the posters use their real names, and are in fact practicing attorneys with subject-matter expertise, some are even pretty well-known attorneys. I post there occasionally under my real name, and on topics that touch on my actual areas of practice.
"What is the significance of this to allowing Sharia Law to determine contract compliance?"
Because the court isn't per se allowing "sharia law to determine contract compliance", it's allowing the contractees to establish the choice of law (and the court of arbitration) that will be used to resolve any contract disputes. In this case, it just happens to be a Sharia court applying sharia law.
"couldn't the whole thing still be challenged by the loser under Article VI"
No. Again, you have to (intellectually) replace the phrase "sharia court" with the phrase "arbiter". The type of arbiter, and the law that the arbiter is applying is (mostly) irrelevant, so long as both parties entered the agreement not under duress or coercion - standards that would apply to any contract.
I would add that arbitration law is complex (and it's not my area of practice). And, it continues to be evolving area of law, even absent instances of Sharia application. Even in this term, the Court heard a case about arbitration where many people were weary of some of the implications (and erosion) of our standards of federalism and other principles of law. That case is AT&T Mobility v. Concepcion.
Lastly, as some other mentioned, there are limits (under the Constitution and other US law) what can actually be contracted. And, the arbiter could not order something that is in violation of US Code or Constitutional Law - like the removal of appendages, or the forfeiture of children.
Some of the hostility to this practice is a bit strange. As I said in an earlier posting, these kinds of arbitration agreements have been going on for decades and decades, and really are quite standard fair in contracts of commerce between international parties.
-PJ
So what if a federal law was adopted outlawing the use of arbitration based on religious law?