Most Mongols were not Christians, but some tribes were "Nestorian" Christians (though I'm not sure that's what they considered themselves? Maybe that's just what Europeans called them?), and a number of influential princesses were Christian.
What was this Christianity actually like, as practiced by the Mongols? Did they have a liturgy, and what language was it in? Syriac? Mongol?
Were there famous Mongol saints they venerated?
Did they produce Bibles in Mongolian?
Did they have Churches?
What was the position of priests and Bishops in the culture of the Mongol tribes that practiced it?
Do we know anything about their theological positions?
I'm also interested in Central Asian tribes that held important positions in Mongolian culture, but were actually Turkic rather than Mongolian. I know that the Mongols sent a Uigher bishop to Rome, so clearly they had Bishops.
This is a vast array of questions - so I'll try to keep my answers brief.
The first thing to note is that there both Syriac and Roman Catholicism existed in the Mongol Empire. Roman Catholicism being spread by European visitors to the region, but ultimately never gaining a large number of converts. Your question, however, seems to be more concerned with Syriac Christians.
Europeans long referred to Syriac Christianity as "Nestorian" although this label has long been rejected. I am unsure how the Mongols referred to the Christianity practiced amongst them, but the Mongols in China, for example, adopted Chinese vocabulary to refer to Christianity.
Scholars such as Jack Weatherford have argued that there were no Churches, no liturgy, and that the religion was based around wandering monks. However, other scholars have challenged Weatherford's thesis noting that contemporaneous European sources refer to the presence of Churches, and pointing to archaeological finds that suggest other Churches existed. A case has also been made for Ger-based Churches. Indeed, looking at the historical record it would appear that Churches existed. Furthermore, since Christian Mongols belonged to the Church of the East, they would have used the East Syriac Rite in their liturgy. Syriac Christianity regularly adopted the languages of the places that it entered so it seems likely that the liturgy would have been in a language accessible to the religion's adherents.
There do not appear to have been any Mongol saints. It is difficult to say whether or not a Bible existed, some have claimed to have translated the text into Mongolian such as the Catholic Giovanni da Montecorvino in 1305CE however no copies of Catholic or Syriac versions in Mongolian exist. The earliest existent translation is from 1827CE.
I don't know much about the place of priests - although the early Mongol rulers patronized the clergy from all religions, and they would have been treated on par with the leaders of other religions. Those sent abroad obviously had quite a high social status since they were acting as diplomats on behalf of the Mongols.
Theological positions would have been consistent with that of the Church of the East at the time.
Sources:
Li, East-Syriac Christianity in Mongol-Yuan China.
Gillman and Klimkeit, Christians in Asia before 1500.
Rossabi, Voyager from Xanadu.
Weatherford, Genghis Khan and the Making of the Modern World.
Wallis Budge, The Monks of Kublai Khan Emperor of China.
Winkler and Baum, The Church of the East.