Response time!
----To Dailato
Currently there are Terrun cardinals and other people of authority on the new continent, but the pope equivalent is still back in Acaelia. Communication between the various temples is facilitated largely through letters, and it remains so for the temples on the new continents. As people continue to filter over to the new land, religiously affiliated people will often traffic letters to the new lands, and the ships that transport people will traffic letters back to Acaelia.
As with any religion, every congregation within it will be a little bit different. Each temple is typically headed by a single person called an Atriarch and most temple's ideologies will be shaped to some degree by their Atriarch's particular views. But there isn't a division ala modern Christianity where there is a variety of named sects, like Catholicism, Lutherans, etc.
As to the owning lands, I'm unfortunately not really able to answer as to that, as I don't want to say anything about in game mechanics I'm not personally sure of.
As to the question about "things made from natural stone only", they far prefer to use things made of natural stone in the creation of their temples. For instance, you would be far more likely to see a stone brazier as a source of light, as opposed to a copper or brass one. The Terrun order will use man made tools, just typically anything that is considered a "holy implement" will be naturally derived, but they are not opposed to shaping stones and rocks with tools.
The way the church is involved with politics is complicated in nature, but for the most part, they have a fairly strong influence. While the church is distinct from the government itself, Terrus is the official religion of Acaelia, and the government will sometimes call in favors from the church and vice versa. But in general they are separate entities.
They don't have a problem with mining at all, in fact they view getting further into the earth as a holy act. So mining is looked upon pretty favorably
----To Elysana
The system of Acaelia is closer to that of a meritocracy, but it still has some bloodline privilege leanings. The closest example I can think of off the top of my head would be the Song dynasty, wherein people had familial privilege but they could rise through the ranks to become very high ranking bureaucrats and aristocracy.
As to your question on gender relations, in Acaelian society women are seen as much more equal to men then in other traditional fantasy games. That's not to say there's no such idea as femininity or masculinity, but in general women can hold any office, and armies recruit men and women alike.
To answer your next question, while some in the aristocracy might have that expectation, warrior women, women who own stores, and very high ranking political officials who are women are not at all uncommon. So while some highborn people may have that expectation, plenty of people in the society at large reject that idea. Additionally, Acaelian women can and do go to war. Some female warriors are going to be included in one the next lore stories, so look out for them
Thanks for your interest guys
May the stones weigh heavier upon your enemies than upon yourself.
Billus