Okay, in the spirit of
suggestions, I offer the following.
I'm going to go through this backwards, just to make sure everything is dealt with properly. My only major gripe is down at the bottom.
(1) In South America, they already had well-developed bronze tools and weapons. The main problem was that they weren't made ubiquitous until the Inca performed a mass Standardization / internal deportation across the Andes.
(2) Uh... what kind of disease? I know there have been 500 years of extra contact between Americans and all the various Old World vectors of disease, but that doesn't seem like enough time to create a disease which is so out-of-left-field for European immune systems. At best, I could see it as yet another version of the plague, and that didn't destroy Europe. Also, if the Europeans got infected with it and suffered, then the minute the Mongols invaded, they should be getting it too with just as much force.
The most I could think of would be syphilis, and that stuff spread rabidly across Europe without blunting its capacity for being the seat of empires.
(3) It's a common myth that Ogedei's death saved Europe, but a quick look at the terrain map of Europe, plus a look at the traits of nomadic empires, disproves it.
The Mongols were a nomadic steppe people, and that kind of people works best on steppe, which can only be found in Pannonia and the Ukraine in Europe. This is why nomadic horse peoples basically stopped in area of Europe: because Germany was too forested and hilly to support a grazing horde.
Any further conquests (ex. China) had to be done with local help (engineers, sappers, bureaucrats, tax men), and this same help had to be relied upon to administer the country because the nomads simply didn't know how and didn't have the numbers to constitute a dedicated bureaucratic class. In China's case, it certainly helped that Mongolia was directly north of China, and any rebellion could be easily responded to.
Mongol conquests were already taking heavy losses at the time of Ogedei's death, and they were getting kicked in the teeth in the forests of Novgorod, Poland and Germany. Even with disease ravaging the areas (which I'm highly skeptical of happening at all), I can only see the areas immediately adjacent to the steppe becoming vassal states. France and Italy are never going to pay tribute to the Khagan.
Hell, just putting New-Old World contact back in the 11th century would already mean that everybody's already got each others' diseases, and the fact that Europeans didn't have the technology or political situation to really colonize at the time means that will be 200-400 years of getting used to the new diseases for the natives without any disturbances. By the invention of the caravel, I'd expect the Americas to be back up to capacity and for certain areas to be more technologically advanced and much more politically unified.
(4) The moment the Mexicans (whichever group they are) got horses, they should've barrelled north like the Furies, because suddenly the wild wastes of Durango and Sinaloa are now prime ranching territory. For more info, look to this video.
They won't extend their reach very far beyond the Rio Grande, but they'll have all of Mexico under their control.
Their population should also just completely explode after this, because the minute they get horses, they get 5x faster transportation of food and a whole load of farming improvements. Historically, the Aztecs had to turn all the land out for a week's walk from Tenochtitlan into farmland and they
still faced perennial famines. With the advent of horses, they'll be able to turn much, much more of the land into farmland, and their population will easily double, if not triple.
This will instantly support an explosion in artisanry and other city-based professions, as more people find themselves capable of packing up and going to the big city to find work. After this, it's almost certain Mesoamerica becomes the seat of a new empire, one based on the millennia-old Mesoamerican sacrifice culture.
(5) Okay, two problems:
a. How did they get across the Atlantic? This isn't a trivial question; the caravel was only developed in the 15th century, and it only
barely made the distance. Roman ships of any era wouldn't make it, and would have more likelihood of reaching Brazil than New England. This isn't just a question of one ship somehow surviving an unnaturally long journey, seeing the land and then somehow making that same journey
except it's longer due to the effects of longitude on the currents; this is a question of people getting there, noticing the land, and then transporting an armada across it.
b.
How the fuck did everybody wait around 8-5 centuries (!!) before some plains tribe got the idea to ride a horse?!!! If horses were around since the Remans established themselves, it's inevitable that some would go feral. IOTL, the Chichimeca of Northern Mexico immediately, and I mean immediately, started riding horses and raiding the Spanish in the standard steppe-nomad manner seen countless times in Eurasia. The Chichimeca did this less than a century after the Spanish arrived. Many formerly-sedentary farming tribes, especially post-Mississippi ones, immediately went nomadic after receiving horses.
For horses to only start to be used for nomadic empire-building in the 14th century, which is anywhere between 8 to 5 centuries after the Romans arrived, the natives would have to be mentally retarded!