If Giant's Grasp leaves the battlefield before its triggered ability resolves, you won't gain control of the target nonland permanent at all.

Gaining control of a nonland permanent doesn't cause you to gain control of any Auras or Equipment attached to it.

The control-changing effect doesn't expire if another player gains control of Giant's Grasp (although this is likely to cause it to leave the battlefield because it can enchant only a Giant controlled by the controller of Giant's Grasp). The effect expires only once Giant's Grasp is no longer on the battlefield.