To summarize before diving into deeper detail, the reason Sherman was publicly derided as insane was that the prevailing attitude in the country at the onset of the war, both North and South, was that the war would be a brief and minor clash. Sherman was one of the few men in the country who understood that the war would be a calamitous one and would require a minimum of 200,000 men to put down the rebellion.

For this figure, he was ridiculed as insane, just as if he had asked for 10 million men, and newspapers of the day rushed to print it as fact, and an ill-advised comment by one of Sherman’s fellow officers was spread across the country as fact before anyone could contradict it. It wasn’t until the battle of Shiloh that people realized how bloody the war was actually going to be, and the events of the war proved Sherman’s estimates in fact had been much too low.

Now, I believe the best source on this is Sherman himself, taken from his memoirs.

He spoke to Secretary of war Cameron in Louisville, KY on what he considered the woeful unpreparedness of the Union line in the center, which Sherman commanded from that state.

He laid out a map of the country in front of Cameron, and showed that in the East, General McClellan was given 100,000 men to hold a line 100 miles across the front. In the far west, Fremont had 60,000 men to hold a line also about 100 miles across. Meanwhile, Sherman had only 18,000 men to hold a line 300 miles across.

He argued that for defense alone, to force the rebels out of Kentucky, he required 60,000 men, and if the Union was to take the offensive, that figure would need to be at least 200,000. This incident was the source for the later claims that he was insane.

As Sherman describes the sequence of events,

“On reaching Washington, Mr. Cameron called on General Thomas, as he himself afterward told me, to submit his memorandum of events during his absence, and in that memorandum was mentioned my insane request for 200,000 men. By some newspaper man this was seen and published, and, before I had the least conception of it, I was universally published throughout the country as “insane, crazy,” etc.”

From there, the rumor twisted and spread across the country as newspapers all rushed to report it as fact, and no one if Sherman’s superiors raised any effort to countermand it.

“About this time my attention was drawn to the publication in all the Eastern papers, which of course was copied to the West, of the report that I was “crazy, insane, and mad,” that “I had demanded two hundred thousand men for the defense of Kentucky;” and the author given for this report was stated to be the Secretary of War himself, Mr. Cameron, who never, to my knowledge, took pains to affirm or deny it.”

Sherman bitterly suffered through these insults for a few months, and he regards how during this time, he resented the suspicion with which his own junior officers regarded his every order as a result. The coming battle of Shiloh however proved Sherman correct in his estimation of the war.

I won’t go too much into the battle itself, but the rebel offensive unfolded exactly as Sherman had feared and predicted. 40,000 confederate troops, under the command of Albert Sydney Johnston, sensed the weakness of the Union line in the center, and fell upon Union forces camped out along the Tennessee River. This had been precisely the portion of the western theater Sherman had held, Tennessee being due south of Kentucky, and had asked to be relieved of commanding as a result of the rumors, and the effect they had both on his own mental health and his confidence in the men under his command to carry out their orders.

Despite the public perception of him, however, which even when he returned to the army saw Sherman placed in rear echelon positions of logistics and supply, Ulysses S. Grant still had faith in Sherman’s abilities as a war commander, and Sherman was placed under his command at Shiloh. The events of Shiloh resulted in over 20,000 casualties and gave Sherman the chance to not only prove his estimation of the war’s requirements in manpower absolutely correct, but also the chance to distinguish himself on the battlefield as a commanding general.

After the battle, both Sherman’s superior officers and the country as a whole realized he had been correct all along, and the war was going to be a long and costly one, redeeming himself in the public eye and causing the rumors to quietly disappear on their own.

Sherman includes in his memoirs a letter written to the public after the war by General Thomas Wood, then in command of the Military District of Vicksburg, regarding the meeting which had prompted the rumors in the first place and the events that followed. Emphasis mine.

“For the purpose of expelling the rebels from Kentucky, General Sherman said that at least sixty thousand soldiers were necessary. Considering that the means of accomplishment must always be proportional to the end to be achieved, and bearing in mind the array of rebel forces then in Kentucky, every sensible man must admit that the estimate of the force given by General Sherman, for driving the rebels out of Kentucky, was a very low one. The truth is that, before the rebels were driven from Kentucky, many more that sixty thousand soldiers were sent into the state.

Ascending from the consideration of the narrow question of the political and military situation in Kentucky, and the extent of the force necessary to redeem the State from rebel thralldom...General Sherman expressed the opinion that, to carry the war to the Gulf of Mexico, and destroy all armed opposition to the government in the entire Mississippi valley, at least two hundred thousand men were absolutely requisite.

…….

A short time after the council was held…an imperfect narrative of it appeared in the New York Tribune. This account announced to the public the conclusion uttered by General Sherman in the council, without giving the reasons on which his conclusions were based. The unfairness of this course to General Sherman needs no comment. All military men were shocked by the gross breach of faith which had been committed.

Th. J. Wood, Major-General Volunteers
Vicksburg, Mississippi, August 24, 1866.”

Source: Sherman, W. T. (1875). Memoirs.

I can't answer the first question, but I will try to answer the second one.

Sherman was assigned command of one brigade which fought at Bull Run. Due to his good performance, he was promoted to commander of Union forces in Kentucky. One of Lincoln's chief objectives in the region was taking East Tennessee, a known cradle of Unionism, with thousands of pro-Union partisans. The administration smuggled arms into the state and encouraged these partisans to rise up against the Confederacy. They did so, burning bridges and taking outposts, but the expected Union reinforcements never came, because Sherman panicked and called off the invasion.

Why did he do this? Sherman suffered what can only be described as a psychological breakdown. He complained of shortages and overestimated the Confederate numbers. He was relieved of command and sent home. There his wife talked of his melancholy and Sherman himself later confessed that he contemplated suicide. This breakdown was deeply humiliating, especially because the press declared him insane and unfit. When Sherman finally felt able to command again he was sent to an obscure post in Missouri. He provided logistical support to Grant during his campaign against Forts Henry and Donelson. Afterwards, he was given command of a division in Grant's Army.

Sherman fought at Shiloh, where he won Grant's trust and friendship, and also a promotion. He was one of Grant's top lieutenants during the campaigns against Corinth and Vicksburg. The pair developed a close relationship, despite accusations against both. Grant was called a drunkard and a failure, Sherman a lunatic. This was the origin of this quote. Despite this, Grant led his army to perhaps the greatest strategic Union victory of the war at Vicksburg, and further victories at Chattanooga. When Grant was called east to assume the post of General in Chief, Sherman was promoted to head of the army of the Tennessee. He won glory for himself in the campaign against Atlanta and his march to the sea.

Basically, Sherman didn't maintain his high-profile. His reputation was destroyed after his failure at Kentucky, and he was only able to build it up thanks to his role in Grant's victories in the west, the influence of his brother the Senator John Sherman and his own later triumphs.

Sources: James McPherson, The Battle Cry of Freedom, and the Memories of both Sherman and Grant.