The controller of the equipped creature may activate the damage ability, not the controller of Heavy Arbalest (if they somehow wind up being different players).

If the equipped creature's ability is activated, that creature is the source of the damage. It doesn't matter if Heavy Arbalest leaves the battlefield or somehow becomes attached to another creature by that time.

The equipped creature is the source of both the damage ability and the resultant damage, not Heavy Arbalest. For example, you can activate the ability to target and deal damage to a creature with protection from artifacts (assuming the equipped creature isn't an artifact itself, of course).

The creature that doesn't untap during its controller's untap step is the one that Heavy Arbalest is equipped to at that time. If a different creature was tapped to activate the ability granted to it by the Arbalest, but the Arbalest is no longer equipping it, it will untap as normal.