Thus, his name was Климент in Russian, so "Klim" is simply the first syllable of his actual name.
So this is more likely an abbreviation than a nickname.
Yes. But I don't know anyone named Kliment in Russian, so I'm not sure if it's an established diminutive, or not.
Sometimes, when there isn't a generally accepted diminutive, people make one up.
I don't know if "Klim" in Russian could be considered the equivalent of "Clem" for Clement, as in English.