Muscovite rulers claimed to be heirs of Rome with the idea Moscow is Third Rome. Not long before that they were a part of the largest land empire ever, Mongolian Empire. Where there any claims to be successors of Genghis Khan? If not then why?

I would assume that their cultural relation to what we call the Byzantine Empire (which was actually the Roman Empire in Late Antiquity and the Middle Ages) and their Christian identity, which was a result of Eastern Roman (Byzantine) emissaries, made them relate with the Byzantines but not the Mongolians which weren't Christian, had a completely different culture -at least from the one the Russian elite had adopted and were conquerors. So they saw Rome (through Byzantium) as their heritage.