Lastly, I have heard things like an MA, MS, MBA, etc. are not helpful for getting into law school. I find this to be a silly conclusion when you look at certain scenarios. Student A holds a 3.9 GPA and 170 LSAT score, but has no graduate studies, nor have they ever lived outside of the general area which they were raised. Student B holds a 3.9 GPA, 169 LSAT score, an MA, and has studied in the Pacific Northwest, the South (North Carolina- boondocks land) and abroad. According to many, the thought proceeds that student A will still be more valuable to the university that both A and B applied to. If it were to come down to only A or only B being accepted, is A really going to win that battle?
Student B is also 22, not that I think that is a very important factor. Obviously, this is all theoretical.
I can't comment on any of the global LSAT administration questions. I imagine the best idea would be to contact LSAC and the testing center directly.
As far as admissions goes, I have a little knowledge, which may not be applicable in all situations. In your example, generally student A and B are never going to be compared directly to each other, except in rare circumstances I'll discuss below. Law schools organize review of applicants a number of ways, but strictly by GPA is not one of them, which is the only way those two candidates are equivalent. They're more likely going to use an index and/or organize by LSAT score. This is because a 3.5 in Basket Weaving from Eastern Middle Central State is only better than a 3.3 in Electrical Engineering from Berkeley for US News purposes. In reality, one is eligible for the patent bar. The only way that Student A and B would be compared together is if they were borderline, either as part of a large group that could go either way, or during a waitlist cycle. During a waitlist cycle the LSAT could be key or irrelevant. If the 170 gets the school's 25th percentile LSAT up or contributes to that, it may be essential. If both are above or below the 25th percentile, it may not even be considered by the school, or it could be determinative. It really depends on what the school is going for at the time.
Most schools will already know whether or not they will admit Candidates A and B based on their numbers. A vast majority of candidates are either accepted or rejected outright, pending a review to make sure no serious criminal convictions or other issues were missed in the acceptance pile and no amazing accomplishments like winning an Olympic medal were missed in the rejection pile. A very small percentage of overall candidates go into the maybe pile, in which candidates can end up accepted, rejected, or waitlisted. As I said above, this is where Candidates A and B could potentially be compared in the same group or even directly if being considered for a waitlist admission.
Now that I've addressed the admissions process generally, I can discuss your question about other graduate degrees. To be honest, most Master's degrees, outside of a technical or hard science field, do not mean much in the legal profession or admissions. It may mean you have the capability to complete graduate work, but if you are admitted to an ABA-accredited school it is already very likely that you have the capacity to succeed in law school and pass a bar exam. I have noticed the JD/MBA combination is a popular route these days.
I did a joint Master's in law school, which I recommend if you can truly get value out of it. I did, but for most attorneys outside of specialized areas of law, they are not that useful. I think the best ones are probably hard sciences, engineering or something that will allow one to take the patent bar, MBA, accounting, tax, or anything finance-related.