Just to make a few things clear, "Great Purge" refers to 1936-1939. The party internal power struggle was mostly in the 1920s. We still often use the term 'purged' what happened in the 1920s and early 1930s but they are distinct period and have very different dynamics.

By late 1920s Stalin was essentially in full control, even by 1925 Stalin was clearly the dominate figure. So the 1936-1939 'great purges' have little to do with power consolidation, and more to do with potentially paranoia, but nobody has really found a way to explain it. By 1931 there is no serous opposition to Stalin, he is by far the dominate figure.

Basically all of the top people that were purged in the 'great purges' were put there by Stalin.

how was Stalin able to avoid this stigma and all of the "people having dirt on everyone else and pointing fingers at one another" phase of it?

Stalin offices controlled the flow of information. They interfaced with the secret policy. They pick who gets what position. His offices had the encryption codes for internal, diplomatic and military communication. They controlled what was published in the party newspaper. They had the right to enforce party rules and punish people who didn't.

Anybody who would seriously speak out against Stalin by 1936 would instantly vanish. Its not even clear how you would do that, there was no independent press. Anti-Stalin articles were often circulated secretly from party member but even that practice became a game with your life as time went on.

Did any of the other leaders try to purge Stalin? Were there any serious plans, or at least trivial considerations, before Stalin consolidated absolute power?

Yes and no. They didn't try to purge Stalin, but in the very early years when Lenin was still alive there were a number of talks that were had by different people that considered removing him from the position, removing the position of general secretary completely or having multiple general secretaries.

The best opportunity was when the so called 'Lenin's Testament' was released. No matter if this document was authentic (there is some debate about that), people then certainly believed it was. A number of the Top Bolsheviks met while on holiday and basically wrote a letter to get some debate going. At that point it does not seem the other top Bolsheviks like Zinoviev, Kamenev clearly did not think they were gone be purged shortly and they did not act decisively.

Other anti-Stalin documents and things were released, but the top Bolsheviks never came together and unified against Stalin. There was certainty a lot of rumblings and behind the back talk but not real attempts to remove him.

Later at the 13th Congress of the Russian Communist Party (1924) Stalin consolidated power considerably, Bukharin, Zinoviev, Kamenev and Rykov and others were essentially all on Stalin side. Stalin basically dominated and after that there was very little chance anybody could remove him.

How did all of the leaders of the revolution and party split off into each of the different factions within the Bolsheviks and Communist Party and so forth, and what characterized their differences? I hope to understand this better, to see how Stalin was able to distinguish and separate himself and achieve total control over the Soviet Union.

As I noted above, Stalin did not simply maneuver his way into power. Lenin created the position of General Secretary for Stalin and gave that position and that office a huge amount of power. So a huge reason why Stalin was successful is because he started out in the prime position once Lenin was out of the picture.

The scope of this question is to large to go into all of the different issues the Soviet Union had and how different people wanted different solutions. Just consider, that the country was coming out of a civil war and had millions of peasants who were not communist at all and were not loyal in any way to the Bolsheviks. Solving all of the problems related to economics, diplomacy, military, party organization and so on created a huge amount of difference, as it does in any state.

To make it short, I would say that Stalin basically positioned himself as the 'true Leninist' and a centrist, while characterizing everybody that was against him as a faction. Again, he was of course helped in this, because he was at the center of power and information.

Only once he was totally dominated, he started to really push his own will in terms of policy.

Would Lenin have led purges to the degree it happened in reality, and would Stalin have been a target?

Lenin certainty played at internal politics and would remove people, but the actual purges from 1937-1939 were a created by Stalin. Lenin would have been very unlikely to do anything on that scale inside the party. Even for Stalin, there is no evidence that the party was not loyal in 1931.

That said, Stalin in his position as General Secretary was certainty the most likely to be able to remove Lenin in the late 1920s, so maybe at some point Lenin would have realized that danger and acted on it. This is of course pure speculation.

In realty Stalin was always loyal to Lenin and executed his policies even when he disagree with them. Stalin for example didn't want to give nationalism so much power and thought it would have been much better to create a greater state, not give Ukraine for example its own republic. Lenin however saw it differently and Stalin as ' People's Commissar for Nationalities' created the ' Union of Soviet Socialist Republics'.