No, powder skis are great for any kind of powder, and they're versatile enough for packed snow too. Race skis are stiffer and have more energy. They are more demanding of good technique and not as forgiving. They're also better on ice. Both have their advantages.
Foam cores are lighter than wood cores, while wood cores give a smoother ride and a more even flex.
A ski containing large amounts of metal may have a tendency to bend, but most skis don't contain enough metal to cause problems. A dampening system is anything that reduces the ski's vibration. Less vibration means better edge grip, more control, and a smoother ride. There are many kinds of systems, all designed to get the same result. The beefy Derbyflex plate is used by World Cup racers.