The middle ability of Enthralling Hold affects only the choice of target as the spell is cast. If the creature becomes untapped before the spell resolves, it still resolves. If a player is allowed to change the spell's target while it's on the stack, they may choose an untapped creature. If you put Enthralling Hold onto the battlefield without casting it, you may attach it to an untapped creature.

Gaining control of a permanent doesn't cause you to gain control of any Auras or Equipment attached to it. They'll remain attached, but an Aura's effect that affects “you” still affects its controller rather than you, the controller of an Equipment can move it during their next main phase, and so on.