Others have touched on this, but ultimately both your examples are based on flawed perspectives. Your question is quite valid, certainly, but both Rome and Temujin were highly predictable outcomes of the existing situation.
In the case of Rome, the Mediterranean had shrunk. Not literally of course, but increasing technology, trade, and sophistication of governance had reduced the effective distance from one point in the Mediterranean to any other. This is why you had Rome and Carthage appearing as powers in the same period and what was (for the period) a rapid succession of wars between them and the Greek states. All the major players had started out small from one perspective or another and they were too logistically close for no one to win. With that in mind the rise of Rome was no more than a mild surprise - the most politically stable player won out.
Nor were the Mongols to be unexpected. The Mongol, Turkic, and Tartar peoples had a military advantage over pretty much everyone in Eurasia in terms of tactics, technology that had no easy answer, and (when within a few hundred miles of the steppes) logistics. I mean, there were Turks of western-Mongolian origin running Egypt at the time. Clearly there was something significant going on. It wasn't inevitable that a competent leader with excellent organizational skills would show up to unify the whole steppe corridor, but neither was it very unlikely - the pressure was in that direction.
As to all the rest, meh.