1. Had Alexander gone west instead of to the east, could he defeated the Roman legions?
2. Were the Romans willing and able to conquer Germania, and would the conquest prevent their decline and fall?
3. Could the Carthaginians build a Mediterranean Empire just like the Romans did in OTL?
4. Was it possible for "Hero's engine" to be developed by the Romans to become a fully-functional steam engine?
5. Could the Celts developed into a unified empire/kingdom in the absence of Rome and repelled the Germanic invasions?
6. Was there any relationship between Christianity and the Decline of Roman Empire?
7. Could the marriage between Charlemagne and Irene unified the Franks and Byzantines?
8. Could the Holy Roman Empire developed into centralized state?
9. Was it possible for the Mongols to conquered Central and Western Europe?
10. Was it possible for the Vikings or Ming Dynasty to discovered the New World and colonized it?
11. Was it possible for Qing Dynasty or Ottoman Empire to industrialized and modernized themselves?
12. Was it possible for Napoleon to won the Napoleonic War and built a lasting empire?
13. Was it possible for the Confederate States of America to won the American Civil War and became a great power on its own?
1) He would have gone for Carthage and Magna Graecia before ever considering Rome, but if he had gone after Rome I'm almost certain he would have succeeded, because bear in mind whilst Pyrrhus was an excellent general he didn't have Alexander's fearsome reputation and he was also prone to becoming distracted very easily. If you faced Alexander, he would keep destroying you until you surrendered or were defeated.
2) Possibly willing, possibly able with sufficient able, but no it would not have saved the Empire; the border would still have had hostile peoples on the other side who would have been displaced in the migrations that took place from the third century AD onwards.
3) Difficult to say; the relative ease with which Spain alone was conquered by the Barcids suggests that the ability to perform large scale conquest and pacification existed. The one disadvantage that the Carthaginians had was that the Greeks considered them their hereditary enemy, and would have reacted to any expansion in the eastern Mediterranean violently.
4) I severely doubt it, Rome already had the capacity to feed a city of a million individuals, many other huge regions and huge populations for the time. When things are already working well it becomes inconceivable for it to become even more efficient. Plus it relies on someone being able to spot the potential for a machine capable of producing independent motion.
5) I'm not sure about unified, it would have required a very lucky and charismatic individual, and those are never certain to appear. Vercingetorix might have managed it had the Romans not conquered him, as he was very much attempting to re-order things along Roman lines in order to fight them.
6) Yes, because it changed the relationship of the Roman hierarchy to the people, it caused inner strife within the Empire as Christianity became imposed on people who wished to keep their previous beliefs, and it affected relationships with nearby peoples in an adverse way.
7) I don't think so, the logistics are just too huge to even comprehend. Having said that, at this point the Franks are not in a feudal system and are more centralised. Whilst unifying them fully would seem impossible some kind of alliance or union would probably not be out the question.
8) With a little more luck, I think so. Also a lot more willpower, because attempts to reform it later in its lifetime were clearly possible.
9) Central Europe sure, but they wouldn't hold it for long. Mongolian rule in Eastern Europe and in many parts of Asia didn't take very long to crumble at all, and the more stretched they are the more likely this is to occur sooner.
10) Vikings would have needed to know to travel further south, and any colony would have had scanty support from the homeland. If they made colonies south before climate change made the far north more inhospitable, it seems possible. As for the Ming, how on earth would they have regularly supplied the West coast of America from the other side of the pacific ocean at all regularly?
11) Ottoman Empire yes, had it been attempted earlier. By the time Westernization was attempted, the Janissary corps was too much of a block for it to properly succeed. As for the Qing, I doubt it. They were far too contemptuous of Europeans given China's history of discovery.
12) Yes, but not by attacking all the places that he did. Probably if he had stuck to Italy and Germany. He might have been able to make a Mediterranean Empire out of all of the various Islands.
13) Bar something shocking, I very much doubt it.