I answered this question on the Mongolian invasion of Europe a few weeks back, which explains why they didn't continue into Europe.

Since this question is slightly different, I'll expand on the old answer by looking at Mongolian military tactics and the significance of Hungary.

Hungary was a friendly environment for the Mongols. It was good for feeding their herds and their horses, and that was a key part of their success in war. The goal had been to capture Hungary, and that was easily done. Following this, Ogedai's death distracted from further incursions to Europe, so guessing what their plans were would mostly be speculation.

However, small parts of it are not speculative. We know that the Mongols fought in inhospitable environments - the invasion of China involved a tumen(army unit of 10,000 men) being sent over mountain passes to attack the Chinese from behind. These passes were considered impassable, but it was done. In Afghanistan, perhaps the most difficult place in history to fight wars, as shown by the long list of failed invasions over the past millenium, the Mongols were again successful.

We also know that the Mongols were masters of long distance communications. Their use of yam riders allowed their armies to keep in touch over huge distances. This was how the Mongolian armies in Hungary knew of the Mongolian victory in Poland within 48 hours, despite the 500km distance. It assured them victory as they knew no reinforcements could arrive to bolster the Hungarian defences. It was also what allowed the Mongolians to scout effectively. When the horde of Batu and Subedei came into Europe, they knew where the enemy armies were to be found, and they knew their strength. They were able to receive constant updates on this, and so outsmarted their opponents.

Knowing this, we can reasonably assume that Hungary would have functioned as a stepping stone to Europe. Whether the Mongolians could have been beaten once they took that step is debatable. With their superior communication lines, and the scouting operations of Subedei and Batu, it's reasonable to assume that the Mongols would not have been defeated. Their use of decoys and ability to trick enemy armies into overextending before turning on them is well known. It was how they defeated the Polish Knights.

So the answer to your question is a bit of both, but mostly on the side of no. The Mongolians were adept at fighting in some of the toughest terrains in Europe and Asia, and had proven this multiple times. The horde in Europe was led by one of the most successful military strategists in history, and it would have been based in a terrain that looked to be custom made for the Mongolian people. It was also supported by extensive intelligence operations, which at the time of their retreat from Europe, had sent scouts into the land surrounding Vienna.

tl;dr The Mongolians had a picture of Europe already forming, and had defeated two large European armies within two days. They were established on the plains of Hungary, and were led by superior commanders. There was no one left to stop them.

You're sort of asking two questions there. The first is whether the Mongols could be defeated in battle and the second is whether they could have dominated the known world. The first question is pretty well covered already by LordHussyPants. The Mongol Army was very well organized and had some of the best generals known to history, any single European kingdom would have struggled to defeat them. Europe was also not known for unifying against common foes so some sort of pan-European defense would seem unlikely.

However, defeating the Europeans in combat is not the same as subjugating the area. Medieval European warfare is full of significant military victories and defeats that led to little to no gain in land. Agincourt is a classic example, no land was gained for England in the wake of that tremendous victory. A big part of this is that Europe was full of fortresses. John Gillingham wrote a very interesting chapter in Maurice Keen's Medieval Warfare on warfare from around this period and one of the things he emphasized was the staggeringly large number of castles in Western Europe at this time. Fortifications were built within a single days march of each other across much of what is now France and Germany. While each one was not necessarily all that impressive the vast number of them combined with the fact that they formed a sort of rudimentary defensive network made conquering anywhere in Western Europe a real problem. While the Mongols had gotten quite good at Siege Warfare by the time they reached Western Europe it was still not exactly their strong suit and the sheer number of prolonged sieges required to take any significant land would have been exhausting. Once you add in the fact that Western Europe had basically nothing the Mongols wanted it becomes increasingly implausible that Genghis or any subsequent Khans, had they lived longer lives, would have had the energy or willingness to take over any significant part of Western Europe let alone to maintain control there.

This is all to say nothing of the problems with conquering Africa, something I'm certainly no expert on. One should point out that the Mongols failed to take over Egypt and not for a lack of trying, like with much of Europe. The Mamluks, led by Baybars, defeated the Mongols at the battle of Ain Jalut, one of the few major defeats the Mongols suffered, which is certainly significant.