How did General Sherman’s views of war compare with those of Abraham Lincoln? How did Sherman’s views compare with those of Colonel Trowbridge?

Sherman, like Grant, did not believe that a war could be successfully prosecuted as a limited war—limited in the sense of not committing full resources to the effort or of not accepting the fact that casualties and destruction of property will probably be large. The problem with Lincoln's commanders in the early phases of the Civil War, such as McClellan, Pope, Burnside, Hooker, and even Meade, was that these men weren't willing to carry out the war in an aggressive manner. They were too cautious, timid, afraid of taking risks and casualties, and they were unwilling to carry the war to the civilian population of the South. The result was that, instead of the relatively quick Union victory hoped for by Lincoln, the war was extended due to the borderline incompetence of most of these generals.
Grant and Sherman understood that in order to win, the principal Confederate armies, those of Lee and Johnston, had to be not merely defeated in battle, but destroyed. The farms of the civilians supplying those armies had to be destroyed as well, and the Southern population had to be demoralized in order for the Confederacy to lose the will to continue their fight. Grant went on the offensive against Lee in the spring of 1864, and despite taking enormous casualties at the Wilderness, Spotsylvania, and Cold Harbor, he refused to retreat, instead moving his army across the James River and trapping Lee at Petersburg. Grant gave orders for the farms of the Shenandoah Valley to be burned, for this was where Lee's supplies were coming from. In the deep South, Sherman repeatedly outmaneuvered the cautious Johnston, took the railway hub of Atlanta, then marched on to the sea, systematically burning farms and property and taking the war to the Southern civilians.
Lincoln has often been portrayed as a man who did not wish to "punish" the South and instead wanted a quick victory that would not disrupt the status quo (including the institution of slavery). There is a grain of truth in this, as his original intention, of course, was not to impose abolition but instead to allow slavery to "die a natural death" by disallowing the spread of slavery to the territories. By autumn of 1862, however, Lincoln realized this plan wasn't workable and that more radical action was needed. He was angry at McClellan after Antietam because Lee had been allowed to get back to Virginia when McClellan had presumably been in a position to destroy his army. Although the Union victory at Gettysburg in July of 1863 was a much more complete victory for the Union than Antietam had been, essentially the same situation occurred after it—Meade allowed Lee to get his army safely through the mountain passes to the west and then back down to Virginia, surviving to fight another day. Lincoln, again, was furious. By this time he realized that men such as Grant and Sherman, who were willing to commit themselves to "total war," were necessary in order to achieve victory.
Your mention of Charles Trowbridge in this question obviously relates to the issue of how the Union forces treated the African Americans who came under the Army's jurisdiction as the war progressed. Trowbridge was in favor of declaring martial law and freeing all enslaved persons in areas held by the Union. This is what Lincoln, unfortunately, had wished to avoid doing in the early phases of the war, as he clung to the notion that the South could be defeated quickly and brought back into the Union with little damage to its existing social structure. However, as stated, by late 1862, Lincoln realized this wasn't possible and that the enslaved people had to be emancipated. Sherman, like many on the Union side, initially held views that were not in favor of abolition, but as was the case with Lincoln, his attitude evolved over the course of the war. African Americans during the war viewed Sherman as their liberator, as his march through Georgia and subsequently through the Carolinas effectively destroyed the system of slavery in those states.

Abraham Lincoln and General Sherman had different views regarding war. Abraham Lincoln would have loved to save the Union without going to war if that would have been possible. However, when that option no longer existed, Abraham Lincoln was fully committed to winning the war. He was frustrated when General McClellan was slow to move his army. However, when the war was winding down, Lincoln knew he had to reunite the country. He didn’t want to inflict unnecessary damage on the South. The peace treaty ending the war was generally an easy treaty on the defeated Confederacy. During the war, President Lincoln was also not afraid to suspend some civil liberties, such as the right of habeas corpus, to win the war.
General Sherman believed in the concept of total war. He wanted to inflict as much damage on the South as possible. He believed this would break the South’s will to fight and would lead to the defeat of the Confederate forces. His march from Atlanta to Savannah is an example of his application of the total war concept. General Sherman didn't always get along well with the press. He also was accused of being insane at times.
Colonel Trowbridge was beloved by his troops. He believed all troops were equal and should be treated as equals. He also had daily devotion with his regiment.
https://www.history.com/topics/american-civil-war/william-t-sherman

https://www.history.com/topics/us-presidents/abraham-lincoln