Did Chinggis really conquer more than Alexander?

No, because the purpose of engaging in a battle is to win it! And Alexander, ultimatelly, won every single one of them, destroying the forces opposing him... if Darius run away instead, that was his problem. There was no organized retreat of any kind, rather a "run for your lives" kind of thing.

Yea, you said Darius run for his life? that means he scared, actually he have nothing to worry about, Alexander almost lost at the same time.

One more thing, Alexander picked a good time for attack Persia when it gets weak. The governers of Persia were corrupt and there sodiers do not want fight for them except for some nobles. Persian sodiers are so bad, they have to hire Greek mercenaries to increase their power and the Greek mercenaries actually helped Alexander. The main reason Persia lost is because Darius escape from the battle(because he just got scared). And they have some problem with their strategy, like they should wait Alexander in Babylon city not fight a battle to lose army. Not sell their ships to cut of Alexander's supporting line, Persia could easy win. How ever, Alexander got %60 lucky to conquer Persia.

Interesting how there are so many shepherds in Greek stories. I wonder if this is the same shepherd that helped Xerxes at Thermopylae. 

" Many books claim that Chinggis conquered China and Persia - but it's not true. Sure most of Chin and the entire Xixia teritorries were conquered, as were Transoxiania and Khwarizm."

Not even, Xi Xia was not conquered until after his death. He only conquered around 1/3 of Jin and many of these cities were retaken, especially after the Jin annihilated the Mongol at Da Yuan Chang.

The extent of Genghis's empire in the U.S. analysis of the top ten conquerors of history was 4,850,000. Hoowever thta undoubtly included Xi Xia and the territory north of the steppe as Temujin already mentioned. Also the Tarim Basin was NOT conquered, the Uighur kingdoms there were independent and only vassal in name. But Genghis's eempireis still larger than Alexander's simply because of the vast steppe territories he conquered.

"nope, but it was larger than Alexanders empire."

No, they are around equal with Alexander at a slight lead with 2,180,000 sq miles and Timur at 2,160,000.

I also don't see a meaning to comparing population at different times, or else, Mao would have been the greatest conquerer of all,

"Other than that, his empire encompassed vast unpopulated areas, while Alexander's empire was of more populated and more civilized areas."

No, by sheer number Genghis conquered far more, the population of Jin on the eve of conquest had roughly 48 million people that does not include Manchuria which has over 2 million population of its own, although Chinngis did not conquer all of Jin he still had at least 15 million people in the territory he conquered. the steppe territory has roughly 1.5 million, the territory conquered of Khariazem had some 4 million people and that of east turkestan at least another milion added up this should be some 22 million in all that he conquered.(although he did significantly reduced them to nearly half that number.)

For Alexander's Empire:

Near East less Arabia: 12.0 million
Egypt: 4 million 
Central Asia and India: 1.5 Million

Greek states: 1.25 million.

All together it couldn't have been over 20 million.