Alexios can attack planeswalkers its owner controls and battles its owner protects.

If Alexios is controlled by a player other than its owner and its controller leaves the game, the effect giving that player control of Alexios ends. Alexios will return to the control of the player still in the game who most recently controlled it.

Alexios can't be sacrificed for any reason. If an effect instructs you to sacrifice it, you can't and it remains on the battlefield. You also can't sacrifice it to pay a cost that requires you to sacrifice a creature.

If Alexios can't attack for any reason (such as being tapped), then it doesn't attack. If there's a cost associated with having it attack, its controller isn't forced to pay that cost, so it doesn't have to attack in that case either.

If Alexios's owner leaves the game, Alexios leaves the game along with them.

If an effect instructs you to sacrifice a creature and you control any creatures other than Alexios, you must sacrifice one of those other creatures. You can't try to sacrifice Alexios.