It is true that the Persians appeared, to Greek eyes, woefully under-armoured. When Herodotus calls them 'gymnetes', he doesn't mean that they were LITERALLY naked - just that in comparison to a Greek hoplite, they might as well have been. In reality, not only did the Persians indeed wear quilted tunics, but many seem to have worn corslets of fish scales - although these may have been confined to the commanders (remember Mardonius' cavalry commander at Plataea). Conversely, to Persian eyes, Greek armies were hilarious cumbersome, easy pickings for their incredibly proficient and mobile cavalry. Don't forget that until Marathon, no Greek army had ever inflicted defeat upon the Persians. Even at Marathon, the key to the Athenian victory appears to have been the absence of the cavalry. Only at Plataea did the Greeks manage to inflict a defeat upon the horsemen of the enemy.

As for the Immortals, yes, the Greeks called the elite of the royal guard the 'melophoroi' - 'the apple bearers'. The dispute about the their more famous title comes from a possible confusion over the Persian word for 'immortal' ('anausa') and the word for 'attendant' ('anusiya'). Most scholars nowadays seem to believe that Herodotus got it right, but there's no way of knowing for sure, of course.