In regard to the idea of the Mongols eradicating Christianity and/or drastically altering European culture, there are some historical nuances that make the topic much more complex. In truth, what makes a good counter-factual stand out from the more, let's call it hyper-focused or Great Man oriented ones, is that they take into account the different social, cultural and political factors of the era in which they're trying to diverge from. Religious history is certainly no exception. We need to the remember that Christianity has always had a number of diverse branches under its umbrella, so to speak. Which can also be said of Judaism and Islam as well.
And while Rome getting sacked and the Pope dying would deal a huge psychological blow to the west, a disaster of that scale would not be unprecedented in the history of the medieval world or the medieval church. There were plenty of instances were things didn't go the way of the Catholic side of the house. Yet its adherents soldiered on and weathered the storm. The Pestilence, more commonly known as the Black Death, dealt a serious blow to the morale of the medieval European Christian, but the faith's practitioners held on. There's plenty of other examples from the history of the church/medieval Europe, but I think the point is made. In fact, it wouldn't totally out of the realm of possiblity for a Mongol-appointed clergyman to become the next Pope and/or for the seat of the Papacy to be moved.
Even early on in Church, a big part of what helped them survive events like the persecution of Decius, was that Rome was already beginning to gravitate towards monotheism in various forms. Just someone wasn't a Christian didn't mean they hated them or cared enough to try and do something about them. More than one Roman emperor regarded them with ambivalence. Sure they weren't defying the Emperor like they should have but that if you were Trajan, you had other things on your mind, besides a religion that was commonly stereotyped as being practiced by old women and slaves. Besides, all they really did was feed beggars and orphans. That seemed pretty harmless to people like Trajan and he wasn't the only one. The point being that religions are often incredible resilient and flexible. If one fades out it's usually because it loses adherents to another practice, more than anything else.
We also can't forget that, as I mentioned earlier, Christianity was never really monolithic. The Nestorians, Eastern Orthodox and Coptic faith are all still around, just to name a few. And two of those doctrines were practiced quite extensively throughout the Mongol Empire. Quite a few Catholic priests also traveled throughout the domain of the Mongols. Though the Mongols dealt harshly with rebellions and were by all accounts, extremely brutal towards the inhabitant of Baghdad, they were generally very tolerant of the local faiths. Sure, churches and clergymen could be and were attacked, but that held true for the Muslim world as well. But Islam would survive the onslaught and even thrive in later years.
While the conquest of the Khwarezmians wasn't a great experience for anyone on the receiving end, it didn't eradicate the regional faith and culture. In fact, attempting to repress local religion is one of the few things that will cause a local people to rise against foreign occupiers. So long as you didn't rebel and didn't make a Mongol commander have to the go the trouble of besieging your city, and you paid tribute on time, they could be very hands off rulers.* Which brings me to my next point, more than one high-ranking Mongol converted to Christianity, or to Islam or to Buddhism. There were even steppe tribes that were largely Christian, well before Chinggis ever showed up on the scene. Kublai is a prime example of this kind of assimilation and far from the only one.
The Mongol Empire was a vast domain that incorporated multiple faiths, languages and cultures under its banner. In fact, as time went on, we don't see the Mongols eradicate the way of life for the peoples they conquered, so much as they themselves eventually assimilate into the beliefs and practices of the people they rule. Again, Kublai is a prime example, but so was pretty much every ruler of the Ilkhanate. And this wasn't unique to the Mongols either, the Germanic tribes that settled within the former western Roman empire did a great deal to preserve Roman law, religious practices and way of life.
After the initial Islamic conquests in the Middle East, who were themselves inclined to tolerate other faiths (for the most part, see the bottom of this post), many of the succeeding Caliphates and Dynasties begin to look very Persian in terms of art, language and culture. Sometimes these were pragmatic measures on the part of the new ruling power, in order to try and legitimize themselves in the eyes of the conquered populace. They could also be genuine gestures and were often both of those things.
So, even if the Mongols do manage to reach the western coasts of Europe, within a few generations they're going to look remarkably like the people they've come to rule and continue to assimilate with them over the next few centuries. Though I have to admit that a Catholic, Latin-speaking, plate-armor wearing, chivalric, Mongol-descended successor/fragment from the old Khanate would be a truly fascinating subject. Though I suppose we'll see, K&G isn't a horrible channel but they've ended up being featured on R/BadHistory more than once and for good reason.
*Obviously there's exceptions, these not hard and fast rules. The Mongols and other conquering groups were not a monolithic cultural group. History is many things but simple is rarely, if ever, one of them.