Some people urged Trotsky to use the Red Army which he commanded to overthrow Stalin and win his fight with him. He refused to do so.
That seems questionable historically. When Trotsky was the People's Comissar of Army and Navy ("Narkomvoenmor") (March 1918-January 1925), there was nothing for Stalin to be "overthrown" from. In those years, Stalin was a very secondary figure - the position of "General Secretary of the Central Committee of the Communist Party" (Gensek) was created specially for him by Zinoviev and Kamenev in 1922, and he remained at that position throughout the 1920-s (see details here, http://www.hrono.info/biograf/trotski.html
). Back then, that position was merely technical: Stalin was indeed a SECRETARY, albeit "glorified" with the adjective "General." His formal responsibilities were, basically, just to keep track of the cadres, sort of like a Human Resources office does it. By creating the Gensek position, Zinoviev and Kamenev were actually trying to put Stalin into a deep shade, themselves keeping much more prestigeous positions and a lot more formal power: Zinoviev was the Chairman of the Communist International (formally the Communist number one in the whole world, the leader of the World Revolution), and Kamenev was the head of the most powerful Moscow Soviet and the Chairman of the People's Economics Council ("Sovnarkhoz").
My grandmother (born in 1909) used to say that when she was a little girl, in 1918-~1922, she and her younger brother used to think that there existed one person with this strange long name, "Leninandtrotsky."
The reason was that these two names were used always together in newspapers and in speeches of various agitators during meetings. So, the kids heard the adults read a newspaper aloud and say, "Leninandtrotsky this," "Leninandtrotsky that." In ~1922=23, however, the name of Trotsky began to disappear from newspapers, controlled by Trotsky's rivals Zinoviev and Kamenev. Yet, Trotsky, as a symbol of the victories of the Red Army in Civil War of 1918-1920, continued to inspire various Bolshevik orators-agitators, and his pictures were everywhere.
The name of Stalin, however, remained absolutely unknown to masses. Again, my grandma said that in December 1929, on Stalin's 50th birthday, newspapers published a greeting and a huge photograph of Stalin on the first page, but many people were wondering, just who in the world was that Georgian with big moustache. Stalin became personally known and popular only in the early 1930-s, especially after the 16th annual Congress of the Communist Party (January 1934), which officially heralded the completion of the first Five-Year Plan. On that Congress, Stalin delivered a speech that contained these very "populistic" words, "comrades, our lives became better, we are now living more merrily."
The Red Army under Trotsky was hardly the same animal as the modern Red Army...
It's really hard to say. Again, in this source, http://www.hrono.info/biograf/trotski.html
, there are many examples of Trotsky's bestial, demonic cruelty. He personally shot delegates from workers or soldiers if he thought that they were rebelious; there are numerous documents certifying that Trotsky personally gave orders to burn whole villages together with their inhabitants, etc.