The triggered ability gained by the creature will trigger when that creature deals any damage, not just combat damage.

Gaining control of a creature doesn’t cause you to gain control of any Auras or Equipment attached to it.

Shackles of Treachery can target any creature, including one that’s already untapped or one that doesn’t have any Equipment attached to it.

If that ability triggers, you must choose a legal target if one is available. It doesn’t matter who controls the Equipment attached to the creature. If that Equipment is no longer attached to the creature as the ability tries to resolve, the ability won’t resolve and the Equipment won’t be destroyed.