Well, when the Mongol Empire breaks up, a new Khanate would be established in Europe. The Mongols would probably put the capital in Hungary, since it is a little slice of grassland in a country full of too many trees for their tastes. They would assimilate to the local culture, as they did everywhere, and model their court off of the European courts of the time.. They would probably use the bureaucratic structures of the Holy Roman Empire to govern the Khanate, and just expand them across Europe. They would recruit Hungarians to act as administrators, since they were very good about co-opting local horse nomads in the places they conquered. There would be greater cultural unity in Europe, and more ready exposure to the technologies and ideas of Asia due to the Pax Mongolica.
They would definitely sack and burn many of Europe's major cities. The Pope, they would probably execute the way they did the Caliph. The Mongols would not tolerate the idea that anybody had more authority than the Great Khan. They would adopt Christianity after a fashion, the same way the Ilkhanate became Muslims, but they would pressure their puppet bishops the organize a more decentralized Church, with a more Caesaropapist bend that put the Great Khan at the top.
Some of Europe would escape of course. England would be safe since the Mongols have a terrible record when it comes to sea invasions. Scandinavia is too cold and distant. Constantinople might be able to hold out, and the Spanish might be able to hold out because of distance, the Pyrenees, and their own legendary cavalry tactics. If so, the remaining free cardinals would probably elect a new Pope in Spain, and condemn the Mongol's pet church as heretical.
In a few generations, the Mongol elite would grow decadent and the Khanate would collapse. England would liberate France and the lowlands (no need for a Hundred Years war, the King of England has the only claim to the throne of France now). Spain would liberate Italy and dominate the Mediterranean, restoring the Pope to Rome. Sweden would take over the old Holy Roman Empire, and the Baltic coast.
And yes, I think you are right. There would be an ultra conservative backlash, just as happened in Russia, in Ming China, and in the Ottoman Empire. There would be no Reformation in this timeline. If the Rennasance happens at all, it might happen in Scandinavia and the British isles, who were far away from both the Mongols and Rome and more disconnected from continental Europe. But it might not even happen there. Japan escaped the Mongol conquest, and nearly kept pace with Europe for years, even independently developing drill-based warfare, before collapsing into ultra-conservatism for other reasons.
I want to modify slightly what you've posted. Don't get me wrong, I fully agree with alot of what you said but I feel like some other considerations should be added:
1) Based on their record of not always needing to invade new territories as merely the stories of their atrocities would be enough to bring about the submission of many neighboring kingdoms who would otherwise face the wrath of the Khans (pun intended for any Trek fans), I would surmise that Poland, Bohemia and Bulgaria would likely have submitted to the new Khanate, especially if a few dozen of their cities were razed to the ground as a demonstration of the fact that despite their disunity, the Mongols were still a force to be reckoned with. The new Khanate might even acquire new vassals such as the Latin Empire of the Straits, the Empire of Nicaea and the Republic of Venice-who would be very eager to gain major economic concessions from the Khanate in exchange for voluntary submission to the Khan's authority,.
2) Continuing with the idea that the Pope would be executed as the Abbasid Caliph had been, the Catholic Church would become very divided as its likely the College of Cardinals would also be executed by the Mongols, and Rome razed to the ground like Baghdad. If Mongol administration were extended into Italy, Rome could be rebuilt as a Mongol administrative capital staffed with a mixture of Mongol and Italian (see Venetian) administrators. This would nevertheless mean that Spain, Portugal, France and England would be left without the guiding spiritual force of a Pope and in the case of Iberia and the British Isles, I would not be surprised if some local 'grass-roots' beliefs or (in the case of Spain and Portugal) a regression back into Islam were to happen. Keep in mind the Reconquista was still far from complete and the borderlands between the Islamic states and the Christian kingdoms was very fluid. A new synthesis of Islam and Christianity could even arise in these borderlands and start to spread across the peninsula. England couild partially revert to the old pagan practices, and France would in all likelihood borrow from Byzantium the idea of their king also being the Head of the Church. This would mean that Catholicism would become a minor, sect religion only practiced by those few states in Germany and Central Europe lucky enough to submit to the Mongols and thus survive.
3) If the Mongols could conquer Tibet, and cross the Caucasus, the Hindu Kush and the Carpathian Mountains, then Scandinavia would not escape, especially as there is a land route for the Mongols to move across. Assuming they do so, Novgorod would likely be subjected to an assault first. Also, with the Pope now executed, its likely that the old Pagan Norse religion might make a comeback in Scandinavia as a result.
4) I'm not so sure the British Isles would fully escape the attention of the Mongols, given that Dover and Calais are much closer than the Korean and Japanese shores. Weather would likely be the one factor the Mongols might still have to contend with, as the English navy was not as powerful as it would later become. Assuming the Mongols took into their service French, Flemish, Dutch and Frisian seamen and shipbuilders, they could in theory construct a fleet of ships capable of carrying a raiding force across the Channel. London would be just as easy a target as Paris, or Kiev. Remember that despite the failure to conquer Japan, Kublai Khan was able to make the attempt in the first place by taking into his service Chinese and Korean shipbuilders and seamen. If he can do it, so could the Khan of Europe.
5) With a Mongol puppet in Poland and Bohemia, Mongols in Hungary and possibly even Italy (alongside the Venetians and any other Italian states that chose the wise course and submitted) the Holy Roman Empire would become a major theater of operations. If a crusade were to be gathered, it would happen here. However, the loss of the Pope could hamper any such calls to crusade, thus the major 'dynasts' of Germany such as the Wittelsbachs, Luxemburgs, Hapsburgs, etc. would likely either have to forge a series of coalitions to at the very least contain the Mongol invasions, attempt to stand against the Mongols alone (and very likely lose) or submit to the Mongols with some chance of survival and likely become vassals, forced to engage against their fellow Germans.
6) No Pope, no Catholic Church. This means no Reformation as there would already be semi-Protestant faiths established on the periphery of Europe, and the Iberian peninsula either fully Islamic once again or under the new syncretic Islamo-Christian faith. Unless France establishes their kings as popes and manages to rally some of the German states to support this new spiritual leader, Christianity is likely doomed as a major force in history. Granted, it would likely survive in the Mongol territories, but it would become one among many faiths and be tolerated only so long as its adherents continued to pledge allegiance to the Great Khan.
7) Lastly, the later Age of Discovery would be very interesting. Assuming for the moment that England's Celtic Christianity manages to survive any possible Mongol raids, they would have a major advantage in the voyages across the Ocean. BUT, this is also assuming that the close proximity to the Mongol Khanate in Europe would be enough of a detrimental factor that no Europeans would be interested in sending ships into the Atlantic, with the possible exception of looking for a way to attack the Mongol flanks, possibly in league with any surviving Islamic forces in the Arabian Sea. Returning to the Atlantic voyages, England could easily settle the Eastern Seaboard of North America or the Caribbean, with their only rivals at first being the Scandinavians. Should Spain and/or Portugal also start sending ships out to explore, they would also become rivals. I would even dare venture to suggest that the Mongols themselves, using navigators from among the Frisian, Dutch and French, could themselves sail west, hoping for a faster way to return to their native fomeland. The discovery of the Americas would thus open a major new arena of competition