If a permanent is tapped, are enchantments on it also tapped?
No, each permanent individually can be tapped or untapped; tapping or untapping a permanent does not tap or untap any enchantments on it. Similarly, tapping or untapping an aura enchantment does not affect the permanent it is enchanting.

110.5. A permanent’s status is its physical state. There are four status categories, each of which has two possible values: tapped/untapped, flipped/unflipped, face up/face down, and phased in/phased out. Each permanent always has one of these values for each of these categories.