If my opponent uses a spell with the effect "return all nonland permanents to their owner's hand", does that include enchantments - auras?
Yes, assuming the enchantment is not also a land, it counts as a nonland permanent. It would be returned to your hand by Coastal Breach, for example.
We know that enchantments go to the battlefield when they resolve.
303.2. When an enchantment spell resolves, its controller puts it onto the battlefield under his or her control.
And finally, we know that any card on the battlefield is a permanent.
110.1. A permanent is a card or token on the battlefield. [...]