From the first link: the blog post refers to interpretation past acts that occurred in foreign jurisdictions, e.g., marriage, divorce, adoption. It does not refer to new acts taking place within the United States, but using the laws of a foreign jurisdiction. Is this correct?
The second link, to me, is essentially similar. Despite the fact that the father took the children to Jordan under false pretenses (a religious divorce), he instead filed for custody in Jordanian Sharia court once in Jordan with his children. The wife traveled back to Massachusetts and filed a case in Massachusetts. Again, an act taking place in a foreign jurisdiction, not in the United States under foreign law.
Regarding the third link, I'm not sure of the relevance. That link tells of the government being or not being forbidden to invest in companies that have some aspect of religious endorsement, such as selling religious wines or clothing or books, when that religious activity is not primary to the business (such as a supermarket selling kosher food amongst all its food offerings). Investing in Ralphs does not endorse Judaism because Ralphs sells Manischewitz. What is the significance of this to allowing Sharia Law to determine contract compliance?
Regarding the fourth link, I suppose the intent was to read the 196 reader comments and not the advert for the radio show. Admitting that I did not read past 5 or 6 replies, my thought regarding building a western barrier against Sharia Law would be this: how does it comply with the Constitution's Supremacy clause, and the 14th amendment's equal protection for all? Even if somebody agreed to Sharia arbitration, couldn't the whole thing still be challenged by the loser under Article VI, "...and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding." To me, Article VI invalidates any Sharia Court ruling, regardless of whether the parties agreed to it or not, if someone else were to challenge it.
-PJ
"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.