From what I studied about the Mongol hold over Russia the Mongol Empire was actually very decentrialized. The khan demanded the swearing of alliegence from all the Russian Princes (then chose the Grand Prince from amongst their ranks), that priests say prays for his health, and a yearly supply of tax. Apart from this the Princes where left too their own devices adn virtually nothing changed. The same old power struggles and feuds still kept going except now they had the Khan to go to too solve major disputes. The Mongol Empire was so succesfull because it was so decentralized by nature allowing its conqured lands to maintain their own culture and too extent former rulers so long as they swore alliegence to the Khan. This worked pretty well as it was like a carrot and stick approach because when vassels got out of line the Golden Horder would make an example of them thus bringing the region into line.

I can not answer all of your questions.

How did the Mongols coordinate and supply fighting wars on multiple different frontiers?

The mongols had a very efficient of communication similar to the 'Pony Express'. This essentially means a lot of stations with fresh horses that can quickly transport messages (or people) from place to place. This means that they could get messages over long distances faster then anybody else until modern times.

Now even this system does not help you to coordinate a war in Europe from Mongolia. The solution for the mongols was essentially decentralization of command. Very early on in the Empire Genghis essentially left one of his generals fighting against the Jin on his own for multitude years. Later his son sent armies in four direction at more or less the same time.

This could only be done if their was a large amount of trust in the general or leader at the front. The armies were usually lead by a combination of loyal generals and members of the royal family. The general knew what was expected of them, and because of the way you became a general, these people new what they were doing. The family members were of different family and the army usually not completely loyal to one branch so that they could not simply take the army.

So essentially you have independent field armies with a lot of autonomy and a ruler who could at least get regular updates on progress.

Now the question of supply. Essentially, because of mongol live style they were relatively well prepared to master that challenge. Their live was not stationary and they could move about much more freely, with all of their wealth. Mongolians had wealth in forms of tens, horses, herds, cloth and such. On large campaigns mongols would take huge herds of animals with them. They had a careful I on were they could go and would prepare their routes with pasture and bridges. The mongols were also great at living of the land. A mongol field army could and did travel around for years without a supply line, they would raid, live of their herds.

Much of what I said here only works for about 2-3 rulers after Genghis. The armies became more and more part of individual family branches domain, instead of a more imperial army.

I've answered your third question in a previous answer regarding the Great Yasa, Taxation, and Communication. Also, this answer regarding food and supply lines should be relevant here.

Some additional background information about Military Organization, the Yam, and language can be found here.

In addition, the administration of the Mongol Empire changed frequently between Genghis Khan and later Khans. For example, when the empire was divided after Genghis' death, each inheritor administered their territory differently. Thomas Allsen backs this up:

Considering the size and cultural diversity of the empire, it is not surprising that the administrative system fashioned by the empreror and his associates was composed of widely diverse elements. The revenue system combined Mongol and central Asian Turkic practices, population registration followed Chinese techniques, and the military machine was patterned after long-established steppe traditions. Although the administrative system had a certain symmetry, it was by no means uniform throughout the empire, particularly at the lower levels.

Allsen provides the example of Möngke continuing a practice of allotting territories and their populations to Mongol princes and other officials. The problem here is that the central government has trouble controlling these offshoots of administrative power.

Though, several sources I've seen say that the entire empire was run according to the Great Yasa, referenced in both answers above. This legal and civil code dictated acceptable behavior for the empire. Allsen also mentions the problem of hereditary succession. He argues that this limits the power of the central government.

I will do another cursory check of my resources, but I believe that this information is still current.

Sources

"Mongol Imperialsm," Thomas Allsen, (Berkley, University of California and University of California Press, 1987), pp. 221-225.

EDIT: Edits are in bold.