*Were
Some examples they gave that come to mind are:
Nomadic people never build large cities, unless you are the mongols
All major empires are agricultural, except for the mongols
No one has conquered Russia or Afghanistan except for the mongols
forcing slaves into your army has never worked, except for the mongols.
There are others, but you get the idea.
The area that we know as Afghanistan has been ruled in part or in whole by various Empires throughout history - various Persian dynasties, Alexander the Great and the Seleucids that succeeded him in that part of the world, the Mauryans of India, the Bactrians, the Kushans, et cetera.
Afghanistan is commonly known as "the graveyard of empires," but I find he less common epithet "the crossroads of empires" to be more accurate. Modern ideas of Afghanistan being an unconquerable for invaders are just that - modern, and driven by the recent experiences of Britain, the USSR, and America.
As /u/LtBobPMonkey points out, the Mongols aren't the only ones to have used slaves in their armies. The most famous slave soldiers in history are probably the Janissaries and mamelukes, neither of which were used by the Mongols. That being said, Janissaries and mamelukes were very different from the sort of slave soldier that I think Green is talking about. I'd have to watch the video, but I suspect that Green is talking about the Mongol's practice of forcing people from cities that they had conquered into military service. These people were basically used as labour and cannon fodder. Mamelukes and Janissaries were not cannon fodder, but rather, were elite soldiers trained from youth, and they often held high positions of importance (the mamelukes basically ran Egypt for a period of time). That being said, children who were sold into slavery and who ended up as soldiers didn't really have a choice in the matter, so they can still be seen as people being forced into a sort of military slavery. The Mongol practice of using conquered peoples as cannon fodder can also be seen as a form of conscription, albeit a very harsh one, and conscription was certainly not a Mongol invention.
Claiming that all major empires were agricultural, except for the Mongols, ignores the variety of other steppe tribes who created their own Empires - the Huns, Kipchaks, Pechenegs, Jurchen, Manchu, et cetera. And while the Mongols themselves were not farmers, their empire came to rule over vast farming societies. And anyways, pastoralism - which the Mongols and those other steppe peoples definitely practiced - is a form of agriculture, but am assuming that Green is separating out pastoralism from sedentary farming.
A lot of this knowledge is really just textbook stuff. If you want a good book on the Mongols, I would recommend The Mongols by David Morgan. I enjoyed it, though I haven't really read any other large books on Mongols specifically.
As for my opinion on the Crash Course World History series, which I have watched in its entirety (albeit a fair while ago), I consider it to be a very general overview of World History that normally isn't completely incorrect, but that really should only be used as an introduction to "better" history. If you show someone who doesn't have any clue about world history the series, they will probably come out of it with a better understanding of history than they had beforehand. I actually think that a lot of Green's material aligns pretty easily with the standard curriculum used in AP World History courses in high schools. A lot of the generalizations that Green makes (ignoring the joke-y "except for the Mongols" ones) are generalizations that align with what is taught in AP World History courses (or, at least, the one I took when I was in HS). I'd even say that Crash Course World History is a pretty good study guide for the AP World History exam. As such, I think it is a useful tool, and I enjoyed watching the series. What Crash Course won't do is provide in-depth analysis or anything, but hey, it's covering world history from the Mesopotamians to the World Wars in a series of Youtube videos.
The Mongols seem exceptional from the perspective of the agrarian empires and states of the Eurasian peripheries that have usually occupied toe historian's focus, which I suspect is what explains Green's comment. However, within the context of central Eurasia they are merely the most succesful of a long series of "steppe empires" whose history stretches back to the Iron Age, and particularly the development of true horse cavalry. The Mongols themselves were in many ways the successors to a long line of Turkic Empires, most notably the Seljuk. You can also go back farther and find comparison to, say, the Huns, Scythians and Xiongnu. Like their predecessors, the Mongols erupted out of their core region due to exceptional martial skill, but also like them, they did not create any sort of persistent centralization in large part because the leadership depended on charismatic authority. A Temujin, Attila or a Chanyu can unite far flung groups and become extremely succesful conquerors through their personal qualities, this did not lead to so-called legal authority. It would be myopic in the extreme to assume that this is a failing of their political system, as they no doubt considered the tedious administrative work a mug's game. However, it did mean that the empires tended to be somewhat ephemeral--the Mongol Empire, for example, lasted around fifty years as a unified entity.
Now, this is not to say that these empires were all alike, and the Mongols, apart from being quite simply the most succesful conquerors in history, also took steps towards administrative centralization with varying success. But generally speaking most of the "exceptions" are not unique to the Mongols, but more the result of the political organization of the central Eurasian steppe empires being very different than the peripheral agrarian ones.
Just off the top of my head:
Nomadic empires did have cities. How you describe large is relative. If your population is smaller than sedentary civilizations, then of course your cities will be smaller as well.
How do you define "major" empire? All empires rely on agriculture of some kind. Nomadic empires frequently raided and traded for products like tea and silk that they did not produce themselves.
Russia didn't exist at the time the Mongols conquered that area. Historians like to make the connection between Kiev and modern Russia, but I don't see how it's any less Ukrainian history or Slavic history. It's a generalization like any other and one that's commonly made because Russia is the modern political entity we most associate with that geographical area. Countries didn't exist in that era.
I'm not sure where you get the idea that Mongols forced slaves into their army. Many of their opponents willingly defected to the Mongols as auxiliaries because it was clear who the winning party was. I've read that the Mongols forced prisoners in front of their army during siege, but that's all I know of slave utilization in the Mongol army. The Mamluks and Janissaries are much more synonymous with what people think of as "slave armies".
Please stop posting answers based on or citing Dan Carlin's Hardcore History. Posts in /r/AskHistorians should be based on academic sources – popular as it may be, Carlin's podcast isn't one.
Nomadic people never build large cities, unless you are the mongols
The Mongols didn't build large cities before Chingis. So what you're seeing is a transition from nomadic life to settled life. This is not a unique or remarkable pattern of history. Continuing to consider them nomadic is to generalize the Mongolian herder of that time to the elite Chingissid court of that time. The two lived very different lives.
All major empires are agricultural, except for the mongols
The Mongols did practice agriculture. Any serious ethnohistory or ethnography of the steppe will tell you as much. Like the Native Americans or Pacific peoples, this was dissimilar to European neat-rows-and-fences agriculture.
No one has conquered Russia or Afghanistan except for the mongols
Alexander occupied Afghanistan. As has another nation, but within the 20 year rule. And what does it mean to "conquer Afghanistan"? Many polities have ruled and been tossed out of Afghanistan, but most of them were Middle Eastern in origin. Are we to consider those an unbroken rule over the region?