So if Xerxes' invasion had been successful, Greece probably would have become a Persian satrapy, right?
Probably...not.
It could have been similar as a mix between the situation of Ionian cities, and the situation in Hellade after the
peace of Antalcidas, after that Attic hegemony was broken and that Persia not only took back Ionian coast, but greatly influenced helladic politics up to the point its gold managed to crush or found foes and allies, or even to threaten the formers of invasions.
Basically, you would end with various Greek states (poleis or basilei) ending up as Persian vassals, with maybe a leading satrap being settled somewhere, but certainly with Greece keeping a large form of autonomy as long they pledged alliegance to the Great King.
Depending of what happens, the situation could last more or less long, ending as an Egyptian analogue culturally (with actual Persian influence but mostly superficial) and as Ionian coast politically; possibly breaking out after a series of rebellions, or ending up as a distinct periphery.
Interestingly again, that would certainly mean even more greek influence on Persian military techniques and cultures (Greek fighters being renowed as mercenaries already IOTL and much used by Persian rulers, governors or claimants)
And if the Carthaginians won the Battle of Himera, Syracuse and probably all of Sicily probably would have fallen under their thumb?
I doubt it : wars, until the Roman era, didn't really involved crushing the opposing side (as the Peloponesian War hints), but more about establishing a domination of the victor, preventing the defeated party to raise again.
If we look at the various Sicilian-Punic wars, which interestingly looks at an ancient equivalent of HYW (a bit like Punic Wars), we more or less see two big sphere of influences fighting on, without decisively managing to expel the other. I'd rather expect a similar outcome to the third Greco-Punic war, meaning Carthage as the dominant power in Sicily (interestingly, with a lesser interest on African hinterland, at least for a while.
I'm not even sure if, save Himera, you'd see much territorial changes even if Carthage won the battle : as much as Greeks didn't really wanted to continue the conflict at this point, Carthagian forces poor supply didn't allowed for a full fledged conquest, would have been it planned.