Interesting questions! And difficult ones to answer. The main problem with researching this area is that we lack any indingeous, Mongol-written source material from this period. Furthermore the histories written by the Mongols' Persian governors, Juvaini and Rashid al-Din, have also ended. As such, we are reliant on sources written by Mongol subjects who often had little interest/contact with the Mongol camps on the steppes.

We thus know very little about any specific Mongol reaction to the plague and any death tolls we do have are anecdotal. There are also difficulties in terms of what the 'plague' actually was. Disease and famine were VERY common in this period. Sources often reference outbreaks of pestilence, but was it plague in the sense of the black death? It is debatable whether there were even plague outbreaks in China at all. McNeill in 'Plagues and Peoples' for example has argued that outbreaks of disease referenced in mid-14th century Chinese sources are outbreaks of the black death. Such would make sense, given that the plague appears to have spread to the West along Mongol trade routes, but we simply do not know.

In regards to how the disease spread throughout the Mongol world, Yersinia pestis, which is the bacteria of the bubonic plague, is transmitted from host-to-host by fleas through blood feeding. The bacteria thrives in the arid conditions of the Eurasian steppes. These fleas are known to have lived on rats, but it's likely that they were also carried by other Mongol livestock such as camels. When these fleas spread to humans, the bacteria is either spread through fleas vomiting in the bloodstream before feeding or through faeces being exposed to abrasions in the skin.

Quite where the mid-14th century outbreak originated from is unclear. The contemporary historian Ibn al-Wardi talks of how it had come from 'the land of Darkness', from which he appears to be referring to inner Asia. The first real records of it affecting Mongols come from the Golden Horde territories in the early 1340s. We then hear or a 'pestilence' in Ilkhanate territories such as Tabriz in 1346. In that same year, Gabriele de'Mussi, a Piacenzan notary, describes how 'countless numbers of Tartars and Saracens were struck down by a mysterious illness which brought sudden death'. His writings are the source of the famous story of the plague hitting the Mongol armies besieging Caffa, with the Mongols catapulting diseased bodies into the city.

Unfortunately, we have nothing pointing to what the situation was like further East. Archaeological evidence suggests that some of the Silk Roads towns were hit, but I am not aware of any evidence about the plague affecting Mongolia itself and, as mentioned, in China it is difficult to discern plague from other outbreaks of famine and disease.

One can fairly assume that death tolls were high. I have never come across any specific mention of numbers, so it is difficult to compare to the impact on sedentary populations. Mongols camps were not as densely populated as cities in Central Asia, China or the Crimea, but then they had more contact with flea-ridden livestock, so it’s hard to tell whether the spread of disease was likely to be any worse or any better amongst the nomads. In relation to how they treated and controlled disease, again I am not aware of any specific mention of plague treatments amongst the nomads. However, we do know a lot about Mongol medicine. William of Rubruck (a western missionary who travelled to the Mongol camps in the 1250s) describes how those who fell sick would be quarantined in their tents, with only their household allowed to enter. It is likely that those who fell ill with plague would have been similarly quarantined. Traditionally, illness was treated amongst the Mongols through shamanic rituals, which could range from exorcism to the burning of animal bones. Medicinal plants were also employed, and on rare occasions patients would be immersed in the entrails of animals. However, the Mongols also borrowed the medicinal expertise of those around them. We know for example that the Ilkhan Hulegu brought Chinese physicians with him to Persia. Ibn Battuta also states that the last Ilkhan, Abu Sa’id, had Indian physicians in his employ. As such, the way in which the Mongols treated the plague is likely to have varied depending upon where precisely in the empire they were, and may well have combined a range of different techniques drawn from Chinese, Arabic and Jewish medicine.

In terms of disposing of the bodies, again there are no specific records relating to the plague. The descriptions of burial in western accounts suggest that the Mongols would simply bury their dead along with their favoured animals and possessions, but whether they opted for other methods such as cremation in face of the spread of disease, I’m not sure.

I hope that goes some way towards answering some of your questions. Given the lack of sources it is hard to discern any ‘cultural shift’ resulting from the plague. However, I think it is worth noting that, whilst the plague coincided with a breakdown in Mongol imperial authority, that much of Eurasia remained in the hands of nomadic groups. Clearly the nomads were not entirely wiped out, and culturally the later nomads under Tamerlane are not far removed from those under Genghis.