Look! Paragraphs!
Wheat is one of the more difficult crops to cultivate and neither it nor oats are all that good for you. Why do you think white (wheat-based) bread was a food of the rich until 1900s or so?
Corn, on the other hand, is easier to cultivate and has notably better hectare yield. The best food crop are potatoes. Why do you think Europeans transplanted American crops on European soil anyway? People didn't like them at all at first.
Maize, as cultivated in South America, was almost inedible without vast amounts of processing--native women spent so much time mashing the stuff that it permanently deformed their skeletons. In fact, if you ate that heirloom corn without processing it, you got pellagra. Wheat, on the other hand, requires relatively little processing--grind it, mix it with water, and you have nutritious dough that won't make you go insane and crap yourself inside out. That's if you had enough to eat at all--the soil was awful, and it was tough enough even living there, let alone getting around to inventing writing. Add that to the crappy choice of pack animals, and the fact that the only really good meat available was guinea pig, and the outcome would have been the same even if you gave those resources to Flavian Rome.
Yes, which is why the single-digit numbers of Spanish cavalrymen did so much damage to five-digit Aztec and Inca armies. The fact they used steel weapons and armor, had guns and carried diseases the natives weren't exposed to had little to do with it, right? Ludicrous. Also notice that the Chinese and many other civilizations had horses.
It was
one of the reasons. The first encounters between these guys read like Crysis 2--the only reason they even managed to put up a fight was how few people the Spanish sent.
Which explains perfectly why this same culture was one of the most backwards places in the world 600 years ago, right? It also explain why it was able to rise so quickly once it adopted foreign crops and importance of agriculture waned. Oh wait it doesn't

I'm not entirely sure what you're saying here, to be honest. I'm talking about two cultures--the First Nations, who had immense resources but no tools to harvest them, and the Chinese, who had cows, horses, AND chickens, but just didn't have the natural resources or chemical adaptation to use them the way we eventually did. I apologize if I was vague.
Yes, the Chinese were quite sophisticated, and even respected by their counterparts on the other side of the continent--but they simply couldn't compete with the multiple transcontinental empires of Europe. They didn't have the resources or the infrastructure--heck, right into the 20th century, whole parts of China were organized legally and literally into hamlets, where people farmed rice with oxen and plows. And now that they've started using our technology, they're one of the most rapidly expanding economies on Earth.
Gauls started out in lush, rolling French countryside, with an added bonus of lush, rolling Spanish and British countrysides, and were still all but wiped out by Rome, a city of mud-bricks they once burned to the ground.
Rome rose because of B). A) was important, but Rome was anything but unique in this regard.
The Gauls and the Celts didn't have the infrastructure to train professional soldiers--mercenary armies may not be the greatest national policy decision from a long-term, integrity-of-the-state perspective, but whatever your views on the primacy of the praetorium or the military-industrial complex, a career soldier fighting in a neo-Marian testudo formation is going to whup any given force of similar size composed of half-naked irregulars. The Gauls were fighting on a level that Rome hadn't bothered with for centuries.
And no, Rome was not unique--in any respect. Any successful imperial power adopts the best tricks from the cultures it defeats--and I have no doubt the Veiites or the Etruscans would have done the same if they'd come out on top. But they didn't, in part because of A. Rome had the same stuff they did, plus a strategic placement that made it a trade hub, plus (possibly, depending on how much you believe Livy) a brief period where the city was a major religious hub and considered more or less sacrosanct.
China had gunpowder 600 years before Europe and outclassed Europe in just about all of those categories ... until 1800 or so. Please explain how come they're only now catching up, without totally negating everything you said.
In fairness, it was a very close race. But this is, seriously, a very good point, and I can't say I have a sure answer. If I had to make a guess, I would probably say for the same reason that Romans lost out to the Goths, and the Byzantines lost out to the Seljuks--they were simply newer and more vigorous, with a momentum that the older empire simply didn't have anymore. Rome completely outclassed every other European civilization for hundreds of years, too--and that, as Humes so helpfully pointed out, neither let them expand beyond the Rhine nor survive the geopolitical ********* that inspired every oh-no-the-monsters-were-just-running-from-SUPER-monsters plot twist ever.
And speaking of China... this is an example of what Sarge is talking about. You know what the English did to them, right? They deliberately went out of their way to get them addicted to Opium, for the express purpose of crippling the nation for their own economic ends--and when they tried to clean up, they went to war with them.
As for size of armies, Incas at Cuzco (1536) had somewhere between 100-200 thousand men. The first time a European nation fielded that many men was during the French revolution, almost 300 years later. A very large percentage of European armies were poorly trained levies, a logical consequence and an important reason for adoption of guns. Aztecs (and many others), on the other hand, had a warrior-based society.
The Gauls had a numerical advantage and a martial culture, too--and Caesar didn't have guns.
Actually, the whole idea of a numerical advantage is something of an illusion--when you consider the number of people working back home to support Rome, even in a rough province like Transalpine Gaul, the barbarians were badly outnumbered. Remember, the Legions had that magic word: Infrastructure. They set out well-supplied, fully trained, and ready to commit themselves completely to the task of kicking serious fundamentum--whereas the Gauls often had their entire society packed into a wagon train not far from the rear lines.