Portugal here, no wood, just iron, steel and concrete. And bricks, of course.

Where I am in South Carolina very few people have basements as they would be prone to flooding

Typically, yes. In the US at least, cinder block houses are common particularly in Florida and coastal regions. The inside would still likely have a moisture barrier and insulation on modern homes, so you will typically have wood framing for interior walls to allow for wiring, plumbing, and insulation that is then drywalled over.

Isn't it way harder to run plumbing/electric through cinder blocks, let alone hanging drywall? Or do you build a cinderblock box first and then frame the inside with wood?

At least in South America (where most buildings are made of brick and mortar) there's no drywall. The internal finish is a smooth layer on top of the bricks and that's it. That makes it easier to hang heavy things on the wall but also makes it impossible to run wires of any kind. It also makes repairs more difficult.

Wouldn’t the inside of your house still have wood framing structure like this though? Extremely rare. Most houses built until the 1959s are made of stone and mortar, with walls at least 2 feet wide. In Northern Europe, i.e. UK, Belgium, Netherlands, parts of France and Germany, you would fund brick and mortar walls. Interior walls were built with thin hollow bricks assembled with mortar. These houses are a pain to renovate and most people just put everything agaisntbthe existing walls and then glue their drywalls on top of it with adhesive mortar. Since then, it is mostly cinder blocks or hollow brichs that are then filled with concrete. Modern building regulations impose steel reinforcement in regions prone to earthquakes. Interior walls are built with a framing of steel railings that are very light and flexible when handled but very sturdy when assembled. The drywall is screwed on each side of these rails which gives room for electrical and plumbung.