I'll try and add my own spin on your initial questions (except the last one, as I don't have anything really worth adding with that), if you don't mind, though it looks like you've gotten some pretty good and quite thorough answers.
The scenario is the Persians win at the Battle of Salamis and the Greco-Persian Wars. The Persians do their thing (Set up some collaborator states, "Pay us tribute, we'll leave you to your own devices"), but many "free" Greeks flee to Magna Graecia. This colony manages to prosper and, once they realize they can't take back Greece proper, just take over the Italian peninsula. After the Persian Empire collapses due to a dynastic struggle a few centuries after the PoD, Greece proper is seen by Magna Graecia as a bunch of traitors and an "eternal" rivalry is born.
The idea of the Greeks flipping out and fleeing to the west to escape Persia is simply probably not going to happen - I'll admit that I've thought that too, but the idea that Herodotus instills, that all the Greeks would get up and leave rather than just live in a Persian occupied Greece because they're "slaves" of Xerxes now instead of "free men" is simply not likely. Most Greeks wouldn't care - and many might even find it to be a golden age, as, after all, there wouldn't be any fighting between the cities, farmers wouldn't have to be carted off to tedious skirmishes, and there'd be more access for Greek merchants with the rest of the world, being yet another part of Persia's empire.
For instance, while some Greeks, notably the Spartans, Thebans, and Demosthenes, weren't big fans of Philip and Alexander's League of Corinth, most were perfectly fine with it, and life went as it did before, except without any real wars which had been almost an annual thing for everyone prior to the League. I wouldn't go as far as to say the League of Corinth was a sort of "Golden Age" in Greece, as Alexander's campaigns demanded a lot of people and there were some rather violent episodes, but life was peaceful and money flowed as well as ever for the merchants, and I'd imagine life under Persia would be about the same, if not better, and so most Greeks would probably accept that as time went on. With Athens gone, I'd imagine that only the Spartans would find Persian occupation absolutely miserable.
Which Greek city-states are most likely to bow down to the Persians? Which are most likely to run off?
Argos is going to be your guaranteed satrapy - it's rich, important, populous, has a ton of religious/historical connections (with the Iliad, various legends, etc.), and - most importantly - actively sought Persia's friendship. In fact, the Argives and Persians decided that they were related through the Argive hero Perseus - the guy who slew Medusa and all that stuff - because Perseus sounds similar to Persia/Parsis/Persians and whatnot.
Thebes and Larissa (the main state of Thessaly) were also not closed to the idea of a Persian takeover, and could be satrapies. It depends on how you want to break up Greece.
Other cities that could bow down... probably none with any real importance. Athens is the main reason for the wars; Sparta's Sparta and has a lot more to lose then most cities with a Persian takeover; Corinth is tied to Sparta via the early Peloponnesian League. Most cities
did bow down to the Persians, just not most of the big important ones.
Is it plausible for Magna Graecia to conquer the Italian peninsula? Is it possible for it to become a power in its own right, like Carthage was?
I'm afraid it's not really probable that Magna Graecia unites, let alone it conquering Italy. While I'm not aware of many action packed wars between say Croton and Taras, they were mostly just a bunch of squabbling cities trying to make money and not get taken over by Italians, not active world conquerers - there are no Syracuses there.
What happens to Carthage? Is it destroyed by the Greeks? Is it bolstered by Greek immigration and prosper?
Carthage is just going to keep fighting against Syracuse in Sicily, likely never actually prevailing over them, if OTL is a good indicator. As Impi says, Carthage was always able to just keep Syracuse at bay, never able to conquer it - largely because of how few Carthaginians there were compared to how many Greeks. A good analog is the French and the British, the British being the Greeks and the French the Carthaginians, right down to the simplified view that Carthaginians having more native allies then the Greeks, and the French more native allies (at least I'm pretty sure they did) then the British - though Carthage did view Sicily as more important then what the French did North America.
You could have Carthage be more lucky in its wars with Syracuse - the Battle of Himera (which incidentally was perhaps the greatest "forgotten" battle of the ancient world, and was fought in the same year as Xerxes' invasion) is an interesting PoD - but it'll be tough to actually get Carthage to conquer Greek Sicily. Similarly, while it's not as hard for the Greeks to conquer Punic Sicily, it'll be hard for them to conquer north Africa unless they have a better leader then Agathocles, which is kind of hard with how Syracuse worked IOTL - tyrants/mercenary captains make for good commanders, not really conquerers. Carthage might have a bit more success then they had IOTL, given that there's probably less chance that figures like Timoleon come over from Greece Proper to disrupt Sicilian wars, but not anything too drastic.
How will "barbarian" tribes' migrations change? Is it even possible to determine this?
I wouldn't worry too much about them... however, as most tribes migrated due to food crises and the like, I'd imagine that'd you see roughly the same tribes migrating.
Where will the Persians go after their victory? Will they be emboldened, or will they stop their expansion? If they continue expanding, where?
See Impi's answer; Herodotus' Xerxes talks of conquering the whole west or something like that, but I've been convinced that Persia's not going to try and conquer somewhere like Greece again - campaigns against some tribes and wars to keep the boundaries where they are, yes, but no real further expansion.
When is it most likely for the Persians to collapse? Which empires would most likely rise from the ashes to regain the glory that was Persia?
With the Greeks out, it's possible Persia makes it a century past Alexander's conquests IOTL, maybe more, but probably less. Its core could last a fairly long time, the core being its territories east of the Euphrates - those are the territories that Persia considered its home provinces, whereas the western stuff are nice additions. For an example IOTL, Darius III (the guy Alexander the Great fought) offered him after Issus all his western territories (and his eldest daughter, I believe) mostly as a way to bribe Alex to just be done with the war.
The western territories will likely just drift away from Persian rule with weaker kings, as what was happening IOTL with Darius and the couple kings before him - they don't necessarily mean the end of the empire though. It's what happens with uber-decentralized empires, like Persia. The core could fall any number of ways - perhaps someone like the Parthians comes along and establishes a Parthian dynasty; perhaps a Western "Alexander" figure comes and destroys the Achaemenids; perhaps they're just overthrown at some point and numerous kingdoms rule the middle east. Who knows?
If you want an "expiration date", if you will, for Persia, I'll say somewhere between 250 and 200 BC - it'd be a Persia without the west, but I can see even a string of incompetent kings lasting with the Iran-Mesopotamia core for a little while.
Hope this helps, and good luck with the timeline!