Do I need a pad under my staple/nail-down flooring?

No, a pad is not recommended for staple/nail installations. However, you should place 15# asphalt saturated felt paper (roofing felt) between the subfloor and hardwood as a vapor retarder. For a cleaner alternative, Roberts SVS (silicone vapor shield) will deliver the same benefits without exposed asphalt or odor and provide a smoother install surface. Do not use red rosin paper, as it provides no moisture protection.