If Willow Satyr stops being tapped before its ability resolves — even if it becomes tapped again right away — you won’t gain control of the targeted creature at all. The same is true if you lose control of Willow Satyr before the ability resolves.

If Willow Satyr leaves the battlefield, you no longer control it, so the duration of its control-change effect ends.

Once the ability resolves, it doesn’t care whether the targeted permanent remains a legal target for the ability, only that it’s on the battlefield. The effect will continue to apply to it even if it gains shroud, stops being a creature, stops being legendary, or would no longer be a legal target for any other reason.