At approximately 38 minutes, Allen is shown reading aloud out of the Book of Mormon. He reads "When the brethren show a unity of purpose, our labor, our toil and anxiety are diminished. Our yoke is made easy and our burden light." This quote is not from the Book of Mormon, but rather The Times and Seasons, a church newspaper published in 1842.
Emma Smith is shown transcribing Joseph's translation of the Book of Mormon, which did actually happen. However, the verse she is shown writing is 1 Nephi 20:10: "For behold, I have refined thee, I have chosen thee in the furnace of affliction" which is not in the part of the Book of Mormon that she transcribed.
Dan (Wyatt Russell) is actually 10 years younger than Ron (Sam Worthington) although Ammon (Christopher Heyerdahl) says Dan is the eldest son.