If you read the reports in this thread, there doesn't seem to be a case of calling the one magic country's call center. It seems to be very much luck of getting an agent at a call center who can see and book the space, which is why people say to call many countries, at various times of day. Sometimes one country seems to be better than others, but it's not clear if that's a pattern or just luck.

Just as a very minor point of terminology, my understanding is that not all connecting flights are "married," that only when sets of connecting flights are treated differently for availability in the GDS are they called "married." I could be wrong, but that's how I'm used to hearing the term used.

It's very common for agents in the US to not see the availability, but there have been reports of success here as well. It's less common for them to see the space but not be able to confirm it. I don't think it's necessarily as simple as "AA agents in the US use Sabre while those overseas use Amadeus," because as you've found, more overseas agents are likewise unable to see inventory than are able to. There's something else, which may be influenced by which GDS and which PoS an agent sets, but seems to be more complex than that.

Yes, but that also means you're competing against far fewer people for those seats, so you're far more likely to get them.

What does EF show? That's what I'd go with. I also suggest setting an EF Equipment Change Alert.