My two cents on the Mongol issue: the heavily-fortified, cavalry-unfriendly nature of Europe alone couldn't stop the Mongols. Look at southern China; it was very heavily fortified and was full of rivers, mountains, forests, rice paddies, and pretty much everything short of tsetse flies that is known to stop cavalry. The Song kept the Mongols from using the rivers for a long time with a powerful river navy, and had a pretty massive army. The fortified cities of Xiangyang and Fancheng helped the Song control the Han River, and they were tough nuts to crack. Thick walls, large garrisons, protected by the river, and they were able to send troops and supplies to each other by boat.
So the Yuan created a navy of their own to cross the river and deny its use to the Song. They used counterweight trebuchets to damage Xianyang's walls and terrorize its people and garrison into submission, and they got Fancheng to surrender by promising its governor the right to rule Xiangyang if he switched sides. Sure, southern China was expansive, heavily fortified, difficult terrain for cavalry. That's why it took the Yuan decades to conquer it, and six years to take Xiangyang and Fancheng. But they adapted and persisted, and used Chinese and Korean infantry very heavily.
Europe, like southern China, was very heavily fortified, and was generally poor terrain for the sort of large-scale cavalry operations that were the Mongols' way of life. Even the Hungarian puszta could only support around 200,000 horses. Considering that each Mongol horseman generally had at least five remounts, this was an issue. Of course, this didn't stop them in China; they just used Chinese infantry as their main force.
I'm not suggesting that a Mongol conquest of Europe was likely. Europe was further away, too fractured for the Mongols to simply make themselves leaders of a pre-existing political structure, and it had a much more militant and aggressive culture than the Song. Any invasion on the scale of the Yuan invasion of the Song would have met a few crusades sent its way, and perhaps Catholics would have been less willing to serve those they saw a foreign heathens than the Chinese, who didn't have the same kind of religious opposition. Europe would have been very difficult to conquer.
Of course, I'm also not saying the Mongols never could've taken over Europe. If they had been as hell-bent on Europe's conquest as that of southern China, and if the Mongols had struck in the days of the khanate's unity, they might have had a chance. Any clever Khan would naturally have played European rivals against each other, promising lords and kings the right to rule new territories and fiefdoms if they sided with the Mongols. Additionally, the Mongols were religiously pretty open-minded, and a clever Khan might have tried to appear interested enough in conversion to win the support of the Pope. It sounds absurd, but Grand Duke Gediminas of Lithuania did exactly that, and got Pope John XXII to occasionally restrain the Teutonic Order, and even won the Pope's blessing for a combined Polish-Lithuanian raid on Brandenburg. Historically, various khans proved quite willing to accept Buddhism and Islam, a fair amount of Mongols were varying types of Christian (mainly Nestorian), and they had pretty good relations with the crusaders in the Levant and the Armenians. If a full-scale invasion of Central Europe were to be followed by the establishment of a breakaway Catholic khanate in Europe, the Mongols could have done pretty well. Of course, if the khan's Catholic and marries European royalty, and the vast majority of the population, soldiery, and nobility are various Europeans, it's debatable whether these conquerors of Europe are really Mongols at all.
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