He took a lot of pictures with Lenin and he used it to convince that Lenin was god-liked and make no mistake. When the picture shows Stalin so close to Lenin, it shows the he's also like Lenin and will make no mistake. This will thus help him earn trust.
When Lenin died, he informed Trotsky that Lenin's funeral was on a later date, however in Russia, they carried out the funeral and Trotsky was seen as an arrogant person and did not attend the funeral. This thus cause more hate on Trotsky. Thus I believe it was because of his cunning manipulation
When Lenin died , he wrote in his testament that Stalin should be removed from all leadership position. When the members hear that, he would surely be removed, however , Trotsky decided not to read that out. Thus, I strongly believe it's his luck that helped him.
Americans wanted to have eternal peace. Out of the 14 Points, we can see that they wanted to have a fair punishment and allow Germany to recover instead of a brutal treaty that force a country to never rise again. Also, America never signed the Treaty of Vasailles thus it was never justified