Hello. Just so you know, we don't really allow hypotheticals. r/HistoryWhatIf is better for that. However, some of what you're asking can probably be addressed without resorting to hypotheticals, so I've gone ahead and approved your question. Things like the usual sentence, etc.

Obviously the answers will need to be somewhat limited, but maybe they will still be helpful. Good luck.

Caveat: I'm a historian of Mormonism, not a legal historian. To answer at least some of your questions:

The charges were very serious and included treason and inciting a riot. Smith and the Nauvoo City Council had the Nauvoo Expositor's printing press destroyed after it ran one issue accusing Smith of polygamy, accusations that were true. (Smith had taken over thirty wives and it is likely he would have also been brought up on some kind of charge relating to polygamy, though there is some debate as to how easily Illinois law would allow for such charges.) The conflict flared quickly as Smith mobilized the Nauvoo Legion and declared martial law. Illinois Governor Thomas Ford negotiated Smith's surrender. He likely would have remained in Carthage for trial and would have almost certainly been found guilty by a non-Mormon jury. It is impossible to speculate on the outcome beyond that. A change of venue to Nauvoo would have been unlikely, but not impossible.

If Smith had been found guilty, he would have done his best to lead the church while in prison, at least for a time. Most of his delegating would have gone to the Quorum of Twelve Apostles. Part of what makes speculation difficult is that Smith often vacillated; there's good evidence he wanted his son to succeed him as leader of the church, but there's also good evidence he meant for the Twelve Apostles to lead. The church was dynamic under Smith, frequently changing. He added to revelations, created new priesthood offices, and revealed new theology. Smith would create an organizational body, give it tremendous meaning and power, and then move on within a short amount of time. At the time of his death, he had created the Council of Fifty, tasked with leading the church in political matters. How it would have figured into a Mormon Church led by Joseph Smith in 1845 is hard to say. He followed the same pattern with people: he would become enamored with someone (Jesse Gause, Sidney Rigdon, John C. Bennett, Amasa Lyman), elevate them very quickly, then have a falling out or decrease their power and influence. Smith's behavior makes it impossible to speculate much on the future of the LDS Church with its leader imprisoned.

On the other hand, it is easier to guess if Smith had been freed or escaped, since he had made his plans somewhat clear. Remember, he was a candidate for US President when he was killed. He would have lost handily and little would have come of it. Joseph Smith wanted to keep Nauvoo as the center of the church, but with satellite branches around the world. This would probably not have happened because, even if Smith had lived, other events would still have unfolded, such as the repeal of the Nauvoo charter in Jan. 1845. Mormons would have almost certainly left Illinois. Before his death there is some evidence that Smith knew that they would have to leave, and he was looking at other possible gathering places for the Mormons, though abandoning the temple would give him pause. He likely saw some future for Nauvoo, but perhaps as a small outpost.

He expressed interest in relocating to Texas, Northern California, or Oregon. Remember that the Treaty of Guadalupe Hidalgo was not signed until 1848; the western part of the continent was Mexican territory. Smith knew he would be leaving US authority and that appealed to him, so Texas is the least likely of the candidates he was exploring. Mormons would probably not have settled in the Great Basin under Smith, though it was also mentioned as a possibility. Northern California and Oregon seem the most likely candidates.

But, and I cannot stress this enough and I think it goes unappreciated by too many historians of Mormonism, Smith was impulsive. His gift was making the world of his followers come alive with religious meaning. On the march to Missouri in 1834 with the Army of Israel (commonly called Zion's Camp), he told those with him that a common Hopewell burial mound contained the bones of Zelph, the white Lamanite who fought under the great prophet Onondagus. He bought Egyptian mummies and said the scrolls contained with them were the Book of Abraham. Mormons could look at the scrolls and handle them. His sermons, his theology, and his community building thrilled his followers. He was, in other words, an enormously gifted leader who could have inspired his people to go nearly anywhere. They could have settled in Kentucky, Canada, Indiana, or Arizona, and Smith would have found a way to tell them that this is where God wanted them to be. People are fascinated with Mormon history because of violence, polygamy, and the like, but it's easy to forget that Mormons were deeply committed followers of God, serving missions without purse or scrip, abandoning homes in Ohio, Missouri, and soon Nauvoo, just to keep that commitment. Joseph Smith was their conduit to God.