If either of Pyromancer Ascension's abilities triggers, it will go on the stack on top of the spell that caused it to trigger. The ability will resolve first. If it's Pyromancer Ascension's second ability that triggered, the copy it creates will also resolve before the original spell. The copy is created even if the original spell has been countered.

If Pyromancer Ascension's first ability triggers, you'll put a quest counter on it even if the card in your graveyard leaves your graveyard by the time the ability resolves.

If you cast an instant or sorcery spell from your graveyard (due to an ability such as retrace, for example), Pyromancer Ascension's first ability won't trigger unless another card with the same name is in your graveyard.

The second ability triggers only if Pyromancer Ascension already has two quest counters on it at the time you cast an instant or sorcery spell. This means a spell can't both cause the second counter to be put on Pyromancer Ascension and become copied.

If you copy a spell, the copy will have the same targets as the spell it's copying unless you choose new ones. You may change any number of the targets, including all of them or none of them.

You may copy an instant or sorcery spell that has no targets.

If the original spell was kicked, the copy is kicked too. Other decisions made for the original spell, such as the value of X or the mode, are also copied.

Both of these abilities trigger when a spell is cast, not when it resolves. This means that their effects will occur even if the Ascension is destroyed in response to such a spell being cast.