Their numbers must have been great to achieve all their victories, I'm curious how the aftermath of their invasion influenced today's demographics in Europe.
Came to an end: do you mean they were stopped, or they just decided not to do it anymore?
The mongols were stopped at Egypt by the Mamluks but decided to stop pillaging into eastern and Western Europe, since the general wealth there was poor compared to the Middle East or Eastern Europe (trade centers such as Riga)
Once they stopped, they generally just maintained their empire. IIRC they tried to romp into Japan but that didn't go over so well. Once Genghis died and it was all split up is where the downfall began. All of the tribes generally went to where they had the closest ties to or where there could be benefits from being in said tribe. For example, if a Uyghur general was granted land, other Uyghurs would go there to be with their own kind.
Genetic traces in Europe? Rough to say. The Crimean Tatars are a sure sign of genetic traces but mainly in purity. I can't say much for there being relation to Western Europeans, maybe in Eastern Europeans. Just a hint... Most of the time, those raping and pillaging stories were propaganda. Genghis embraced them because if you had people making stories about how terrifying your people are, the scared people are more likely to capitulate!