We are putting hardwood floors down. Adding an area rug, is padding a good thing to put underneath, is it necessary?

Not exactly padding but you do need a padded mat. This is about a 1/4" thick pad in a waffle pattern. It's main purpose is to keep the throw rug in place so that it does not slip when you step on it. The pad will also protect the hardwood and give the carpet a little cushion.