Ashley Farr, once Miss North Salt Lake Teen USA, is the first in her family’s long line of Mormon women to become a missionary, and in December she embarked on her new life in this gray corner of Asia. She packed her bag according to the church’s precise instructions: skirts that cover the knee, only one pair of pants, earrings that dangle no longer than one inch, and subtle but flattering makeup, modeled in photos on the church’s website.