I'll speak of the Iberian peninsula first.

The Iberian peninsula was still in the Iron age by the time Phoenician settlers

The Iberian peninsula, in no way, shape, or form, had any contact with Hellenistic states before the Phoenician civilization. In fact, the Iberian peninsula was still in its Bronze age (before 7th century BCE) by the time the Phoenicians founded Cadiz, the first city in western Europe. These ancient cities must have taken a large Phoenician influence even before Carthage entered the scene. Centuries after in the 8th century BCE, Greek colonies started emerging, like Empuries and Saguntum. However, this was much farther north than the Phoenicians had gone. However, fast forward to the 3rd century BCE, Hasdrubal the Fair constructed Qart-Hadasht, known as Carthago Nova to the Romans. We can see Hellenistic architecture used to create the Barcid palace there, making some of the faces with stonework. So we can conclude that originally, Iberia must have had a heavy Phoenician influence; as time went on, Hellenistic ideas and architecture started being added until the Romans took over.

Sicily was quite the same: both the Greeks and the Phoenicians colonized it in the 8th century BCE. The most prominent Greek colony would be Syracuse, while the most prominent Carthaginian colony would be Ziz (Greek Panormos). You can still see the Phoenician presence in the city today, now known as Palermo, with its humongous necropolis for the Siculo-Punics. Carthaginian dominance on the island happened around the same time as the Macedonian Empire, so Hellenistic-style building must have reached Sicily much later.

Carthage itself was a mixing pot: it was a trade port that had a great position in the Mediterranean sea. Carthage's Phoenician culture spread to the Amazigh that lived there originally as well as others. It's not unreasonable to say that it must have taken in Hellenistic, Hellenic, Libyan, Etruscan, and even Iberian influence as well. I could even imagine that the cothon, Carthage's great harbor, may have taken in Hellenistic designs, as it was made after the second Punic war.

So long story short, Carthaginian territories had strong Phoenician influence, but also took in Hellenistic vibes. This criss-crossing of cultures is most evident with Herakles-Melqart, the comparison of the Canaanite god Melqart with the Greek god Herakles, who were both known mythologically for colonizing faraway lands in the west (note the Pillars of Hercules)