No. Brigham Young was in the eastern states.
Joseph Smith was the founder, and he was in upstate New York at the time. Brigham Young took over after Smith's death.
In 1832 when Brigham Young converted to Mormonism and moved to Ohio.
Joseph Smith and Brigham Young
Joseph Smith.
Brigham Young lead them after Smith's death to Utah.
Brigham Young.
No one killed Brigham Young. He died at the age of 76 after his liver ruptured. You may be confusing him with Joseph Smith. Brigham Young took over the presidency of the LDS Church after Joseph Smith was murdered. Joseph Smith was martyred by a mob (shot 4 times) while imprisoned in Liberty Jail, Missouri on unspecified charges.
Brigham Young.
Brigham Young actually didn't make many changes to the Church. Everything that Brigham Young did was in response to previous instruction given by Joseph Smith. For example, Brigham Young moved the Church to Utah because Joseph Smith prophesied in 1842 that the saints would find peace in the Rocky Mountains. Brigham Young expanded the practice of polygamy, but Joseph Smith had preached in favor of the practice as early as 1831 (polygamy officially began in 1843, still with Joseph Smith's approval). Brigham Young also expanded the practice of consecration (the 'United Order'), which Joseph Smith taught in 1834. (The practice is also found in the Bible and the Book of Mormon.) Brigham Young also expanded temple building (begun by Joseph Smith) and performed temple ordinances, which were given to him by Joseph in 1842.
No, nor was he ever accused of doing so. Brigham Young was serving as a missionary in Europe at the time of Joseph Smith's murder so he could not have been involved with the event.
In 1844 the Mormons followed Brigham Young to what would become the Utah Territory.