Shady Spectacles

What happens if I choose to copy an empowered unit when I equip an unempowered unit with Shady Spectacles?

Being empowered is a status and is not copyable, thus the equipped unit will remain unempowered.

Shady Spectacles applies a copy effect to the equipped unit. Copy effects copy the printed (or copied, if another copy effect has already been applied) traits of the unit being copied. empowered is a status, a temporary modification, and not part of the printed traits of the card, so it will not be copied by Shady Spectacles. Whatever temporary modifications are applied to the unit as the copy effect is executed on that unit will be unaffected by the copy effect. If the unit has Might increases or decreases, a buff, the empowered status, Temporary, attacker, defender, etc those will remain in place.

Relevant Rules: 477.1.b.1.b.

What happens when my opponent's Akshan, Mischievous steals my equipped Shady Spectacles? What happens if they want to copy Aphelios?

If it was attached to a unit, the equipped unit loses its copy effect. Akshan will then become a copy of a friendly unit of his controller's choice. If he becomes a copy of Aphelios, his ability will not trigger.

This question deserves a bit more depth. As of the Vendetta rules update, we have clarified how effects like Shady Spectacles and Svellsongur work; they are replacement effects that replace the event of the equipment being attached to a unit with that event plus whatever their effect specifies.

In the case of Svellsongur, the resulting series of events will be:

Attach → Copy the equipped unit's text to Svellsongur's effect text for as long as Svellsongur is attached

For Shady Spectacles, the resulting series of events will be:

Attach → Choose a friendly unit and make the equipped unit a copy of that unit for as long as Shady Spectacles is attached

These events are not simultaneous. The attach event executes first, and then whatever event is instructed by the replacement effect is executed. Thus, the answer in the Spiritforged FAQ is no longer true: if you attach Svellsongur to Aphelios, he will not trigger twice. Svellsongur doesn't have the copied effect text when the attach event executes.

Similarly here, if Akshan nabs Shady Spectacles from an opponent, it will attach and then make him a copy. If his controller chooses to make him a copy of Aphelios, it's too late by that point for his new triggered ability to see the attachment event; it's already been completed.

Relevant Rules: 369.1., 370.1.b.1., 370.1.a.2.

My Sand Soldier has Shady Spectacles equipped and is a copy of Azir, Ascendent. What happens if I activate the copied Azir's activatedability before detaching the Shady Spectacles and then attach them to the same Sand Soldier as a copy of the same Azir, Ascendent? Can I use the activated ability again?

You can.

Each instance of the copied ability is a different ability. The game object remains the same throughout this whole process, so any temporary modifications will remain on the Sand Soldier as it transforms, but the copied rules text is an entirely new instance of the ability. It is the same idea as Svellsongur's copy of an ability being "refreshed" if Svellsongur is detached and then reattached to the same unit.