Atemsis, All-Seeing and Mask of Immolation interaction

Question

Does sacrificing Atemsis, All-Seeing equipped with Mask of Immolation trigger Atemsis' second ability, or does it not trigger because Atemsis left the battlefield already?

Atemsis' second ability:

Whenever Atemsis, All-Seeing deals damage to an opponent, you may reveal your hand. If cards with at least six different converted mana costs are revealed this way, that player loses the game.

Mask of Immolation's sacrifice ability:

Equipped creature has “Sacrifice this creature: It deals 1 damage to any target.”

(Emphasis mine)

Answer 1

No, it doesn't. While it's certainly Atemsis, All-Seeing who does the damage (so a Circle of Protection: Blue would work to prevent it), the damage is done at a moment when Atemsis, All-Seeing is in the graveyard (it has been sacrificed as part of activating the ability). It has the reveal-your-hand ability only when it's on the battlefield, so it won't trigger.

113.6. Abilities of an instant or sorcery spell usually function only while that object is on the stack. Abilities of all other objects usually function only while that object is on the battlefield.

A list of exceptions follows but this ability is not one of them.

Answer 2

No.

What matters is the state of the board at the moment that the damage was dealt:

603.10. Normally, objects that exist immediately after an event are checked to see if the event matched any trigger conditions, and continuous effects that exist at that time are used to determine what the trigger conditions are and what the objects involved in the event look like. However, some triggered abilities are exceptions to this rule; the game “looks back in time” to determine if those abilities trigger, using the existence of those abilities and the appearance of objects immediately prior to the event. The list of exceptions is as follows:

None of the exceptions apply to things that trigger upon damage being dealt.

This means that when damage is dealt, the game looks at the state of the board to see if there are any triggered abilities that trigger. And as per rule 113.6 detailed in Glorfindel's answer, Atemsis is in the graveyard at that time and thus its ability does not function.