@ToyoRai: I imagine the conversation was something like Byakuren asking if Koishi if she wanted to join the temple, and after a back and forth of getting nowhere, Byakuren, strongly implied, that Koishi already wanted to join the temple, at which point Koishi, having no actual opinions of her own accepted the outside suggestion, and went along with Byakuren.
My thought on the layperson thing, where else is Koishi honestly living? Along with the slight foul play I just accused Byakuren of, she probably also added that Koishi lives at the Myouren Temple. Koishi, following the same logic as above, agreed.
I dunno, I don't want Koishi to be a Buddhist or anything really. Her story (as I know it) is bittersweet and wrapped up nicely, but I'll try to keep an open mind (also Byakuren might not be as bad as I'm making her out to be).
It makes sense that of all people in SA she'd be involved then.
Okuu has a god inside of her. Maybe she needs to gather faith too... or does she (seriously, have you noticed that the Yaggaratsu in Okuu is a god with seemingly no worshipers, and yet is constantly powering the fires of hell).