It's the same for Taoism, actually. The Chinese would flock to temples or shrines where certain gods, or aspects of gods are known to be particularly effective at "answering" prayers. Needless to say, the said temples would grow in prestige and wealth, and this in turn draws even more worshippers, further boosting the gods' "influence".
The key difference to note is that temples and shrines don't have "congregations" the way Westerners understand the word, ie, a group of worshipers who dedicate themselves to "one true god". Rather, as I explained above, the Chinese have a very pragmatic approach to religion: Everything is seen as a transaction. The more one offers to the god or gods, the more one can expect to get in return (and if one gets nothing, well, it's perfectly understandable that one would turn to other gods for help). As far as I know, the Japanese approach Shintoism in much the same way.
I'm not too sure about that. It's more accurate to say that the Scions are agnostic. They acknowledge the existence of gods pretty much the way all Eorzeans acknowledge the existence of aether. It's simply a fact of life. And the gods, and their powers, are to be tapped the way they would any other tool (hence Louisoix' plan to use a fraction of the Twelve's power to cage Bahamut).
Moreover, throughout this time, I don't recall any Scion ever praying to Hydaelyn the way they did that fateful day before the Calamity. So, in a sense, they acknowledge Her existence, but that in itself is not remarkable enough to merit "worship", as far as they're concerned. It's more like they understand that Hydaelyn is the very embodiment of their world, and if She dies, so does their star.
And that is the fundamental mystery of Eorzean religion.
To ask if the Twelve exist is very much the same as asking whether God exists in our reality. No one knows. The Scions, like all Eorzeans, simply have faith that They do.