Genghis Khan is said to have worshiped "The Eternal Blue Sky." One story I read said he made offerings to a mountain god. He showed respect for the churches, mosques, and synagogs of the peoples he conquered. How religious or spiritual was he and what were the tenants of his faith?

I'm going off an older source, so please correct me if you are aware of a more modern interpretation of the record. There is definitely plenty of room to expand on this answer, but I wanted to get something down now that everything as been deleted.

Many Shamanist Mongols at the time followed what is now referred to as Tengrism (sometimes Tengriism). There are many different subsets of beliefs within this umbrella term for Central Asian religions of the period. Polytheism was common. Temuchin, Genghis Khan, is thought to have worshipped Tengri as the leader of a pantheon of Gods. Tengri was the "Sky-Father," and Temuchin and his followers crafted a legend that he was called upon by Tengri to conquer all peoples.

It is unknown exactly how his spirituality was influenced by his contact with different religious groups during his conquest. He remained a deist, and he was not a Christian or Muslim. Any reportings surrounding the evolution of his beliefs after meeting the Nestorians and Muslims is nigh impossible to verify or separate from fiction.

Earthy, E. D. (1955). The Religion of Genghis Khan (AD 1162-1227). Numen, 228-232.