A lot of this is explained in the game, but here goes:

You're right, Dream Zanarkand has nothing to do with Sin. The fayth and Yu Yevon are what caused the Dream. Zanarkands survivors of the Machina war knew their city was doomed, so became the fayth on gagzet. They are not quite dead, but also not alive as far as I understand, but the fayth are people whose bodies have turned to stone, and whose soul is trapped.

Sin was created as an armour to protect Yu Yevon as he summoned from the fayth what is known as Dream Zanarkand. It is considered a dream I think because it is the dreams of the fayth that make it possible, and after 1000 years, you cant blame them for being tired.

The Yevon religion is told by maechen to be the result of a deal between Bevelle and Lady Yunalesca. She defeats sin and he gets eternal glory. It is also said to be a possibe way of lessening Sin's damage, as bevelle originally thought that Sin was being Summoned by Yu Yevon with the fayth on gagazet, so they wanted to try and suck up to him so he wouldnt set sin on them.

As for the machina, I dunno, It might be an attempt to try and prevent another machina war...