Akshan, Mischievous

Q: I play Akshan paying its additional cost. The play effect (1) to “steal” a gear is finalized on the chain targeting opponent’s Gear A; in response, I play Thrill of the Hunt, targeting my Akshan, and the spell resolves. Akshan is played again paying its additional cost, and the play effect (2) to “steal” is finalized on the chain targeting opponent’s Gear B. Play effects are now on the chain in the order 1 > 2. What happens to Gear A when the first Play effect resolves?

A: Control of it will not change.

As a premise: the Akshan played via Thrill of the Hunt is considered as a different game object than the first Akshan, as it changed zone to a non-board Zone. Let’s say the second Play effect resolves (control of Gear B is gained) and the first is now resolving: Gear A moves to my base; Akshan’s ability reads “if it’s an equipment, attach it to me”, so if Gear A is an equipment, the cited instruction fails to find “me” since the Akshan that was the source of the ability that moved Gear A is no longer on the board, therefore it doesn’t attach; after the resolution of the ability, a cleanup will occur, in which Gear A will move to my opponent’s base again.

Relevant rules:

• 359.3.e.4. If a target changes Zones to or from a Non-Board Zone and then returns to its original zone, it is no longer a legal target, because it's not treated as the same object. [This rule can be found in the Targeting section, but its logic is applied also outside of targeting rules].
• 323.7. 5. Recall all Unattached non-Unit Gear at Battlefields, and all Permanents and Runes in Bases other than their controller’s. Remove all Hidden cards from all Battlefields that are not controlled by the same player and place them in their owner's Trash.

Q. If I take control of an equipment then move the equipment to another unit (say with Jax's ability for instance), what happens when Akshan dies?

The control of the stolen equipment is regained by the original controller, but the equipment remains attached to the unit until it dies or the gear is moved/detached by another ability.

718.5.e. Attached cards may have different Controllers from their Top-Most card.
718.5.f. Changes in Control of the Top-Most card do not impact Control of Attached cards and vice versa.
718.5.g. An Attached card still appends the abilities in its Effect Text to the Rules Text of the Top-Most card and modulates the Top-Most Card’s Might by its Might Bonus.