Can't speak about Italy or Spain, but the Persians definitely had known about Carthage because the people who founded Carthage were from Trye. Trye was in Persia until Alexander slaughtered most the city, save the people that were in a temple.

I can't answer the former but almost certainly the latter. There was a tremendous amount of trade going on within the Mediterranean. Spain was a major source for metals and its existence most certainly would have been known. Italy had both a major Greek colony as well as an enormous amount of trade with Greece (most surviving Attic vases come from Etruria). As the Persians had contact with Greece, either from the Ionian Greeks or through vassals like Thessaly, this sort of intelligence would almost certainly have made its way back to Persia. Others have mentioned that Carthage was a Phoenician colony and the Persians had conquered the Levant.

Two questions:

What Persia are you asking about? As far as Achaemenid (550 BC)? Although I am certain Khosrow II would know about these countries :P

What Carthage and during are you talking about? Carthage before or after the Punic wars?

A basic answer without much data would be - Yes, they knew about Italy, they knew about Egypt (which is north African) and her neighbors. I am not certain if they knew about Spain/Iberia.