As you admit, Europe had people. Plenty of people. That's a huge military and economic advantage against an invader that absolutely cannot move large numbers into the territory.
Europe didn't have especially large populations. And besides, the Mongols would have brought the plague with them, which would have wiped out a great deal of the opposition. Of course, it would have effected the Mongols as well, but less so as they knew how to avoid it.
There simply was no way for the mongols to transfer large armies across the wastelands of Asia to attack Europe, even if they did tried to leverage on their chinese resources.
They wouldn't really need to. When Tamerlane conquered his massive Empire and wiped the floor with all the great powers surrounding him, he didn't have the resources of China with him, nor did the Mughals when they descended upon India. All of them fought as the Mongols did, with overwhelming Nomadic military superiority.
They'd have to conquer and submit large numbers of europeans, then use them to attack the rest of Europe. It was not going to happen: they never even managed to subdue the princes of the Rus, only get the occasional tribute mixed with rebellions.
No. The Mongols subdued the Rus without a doubt. If you read the history of the Mongol invasion of the Rus, it is a very onesided war in which the Mongols effortlessly destroyed one army after another and then proceeded to sack the cities. Pskov and Novgorod were the only major cities to avoid sacking.
The legacy of the Mongol yoke shaped Russia politically, especially Muscowy, possibly as much as the Byzantine legacy shaped Russia in culture. The Golden Horde was known as the Golden horde because of the amount of funds they managed to extract from their subjects, which was fortune. This was hardly an "occasional" tribute that the Rus paid when they felt like it.
The reason why the Mongols never simply conquered the Rus and subjugate them directly is probably a combination of the lack of local centralized administration (present in China and Persia) and the Mongol's own lack of interest in administrative duties. The Mongols, especially the Golden Horde, always saw themselves as a predatory elite, who sought to extract what they wanted from the weaker peoples. This predatory philosophy culminated in the Empire of Tamerlane, whose idea of empire was to attack everyone, steal whatever they've got and bring it to his capital in Samarkand. This was more or less the reason for their religious liberalism: they really didn't care about religious issues.
Incidentally, it was Tamerlane who broke the back of the Golden Horde. Not Russians.
Europe, unlike northern China, was too fragmented and too foreign to be conquered and then controlled by the mongols. The mongols who did get there across Asia as reinforcements would be constantly ground down fighting rebellions.
Political fragmentation is a weakness, not strength in this discussion. Political fragmentation would have meant that the Mongols could have just dealt with them more easily, especially in the short term, which is important in the question whether they can conquer them. In discussion of long term rule, political fragmentation is a definite disadvantage for the Mongols and yes, this would have made a Mongol yoke in Germany fairly short-lived.
The mongols, like the Romans, the Arabs, or the Ottomans at the maximum extent of their empires, were pushing the limits of what with their technology could be controlled by a single imperial center, or even a single people, without losing its cohesiveness, without losing the ability of suppression rebellions. Their limits were larger because their mobility (nomads on horseback -> faster communications) was greater than the military of those other empires, but they had reached them in eastern Europe, in the territory of what became the Golden Horde.
The Mongol Empire was not a single entity. It was composed of many Mongol Empires from the start of its greater Asian extension. The Ilkhanate and the other Khanates were created with that understanding more or less obvious to the Mongols. Even so, there is no reason to presume that the Mongol Empire was over-extended. It had a very fast system of internal communication: a Khan in Moghulistan would have been informed of the events in Germany faster than the Roman Emperor in the Persian front was informed of the Gothic invasion of Thrace.
In the end, it was the fragile tribal nature of the Mongol empire that was the reason for its stagnation and disintegration, not vague reasons of over-extension.
It is proof of their military success that they did reach those limits (not that I'm accepting that the mongols would have fared well fighting in Germany, for example), but logistics were always the ultimate limitation on empires with successful armies.
Logistically, there is nothing preventing the Mongols from moving a large army from China to Russia and then further west. It is once they enter hostile territory that dangers of attrition really begin. Germany is not that different from China, so I don't see why they would not have wiped the floor with the small disorganized German bands that Europe called "armies". Mongols would not have been the first steppe nomads to dominate Germany (Huns did it before them with far less resources, and Germania was not empty woodland as is mistakenly believed).
