On opponents turn I chump blocked to trigger Archangel Avacyn's transform on my upkeep.
Opponent has Giltspire Avenger on the battlefield, and Faith's Shield in hand. Is it possible for him to cast it for pro-red, before Avacyn's transform into Purifier deals it lethal damage?
If so and even if not so, does the 3 damage Avacyn does to opponent allow Giltspire to tap and destroy it?
Yes, your opponent can do both of those things and destroy Avacyn, unless they have less than 5 life, in which case no damage will be dealt.
First, Avacyn's ability normally plays out like this, if the opponent does nothing.
At the beginning of the upkeep, Archangel Avacyn's triggered ability goes on the stack. At this point, both players have a chance to respond.
Archangel Avacyn's ability resolves, and she transforms into Avacyn, the Purifier. Avacyn, the Purifier's triggered ability goes on the stack. At this point, both players have another chance to respond.
Avacyn, the Purifier's ability resolves, and she deals 3 damage to each other creature and each opponent.
Your opponent can cast Faith's Shield after step 1 or 2 to save the Giltspire Avenger.
Assuming the opponent acts after step 2 and they have more than 5 life, the full scenario you described plays out like this:
At the beginning of the upkeep, Archangel Avacyn's triggered ability goes on the stack.
Archangel Avacyn's ability resolves, and she transforms into Avacyn, the Purifier. Avacyn, the Purifier's triggered ability goes on the stack.
In response, the opponent casts Faith's Shield targeting their Giltspire Avenger.
Faith's Shield resolves. Giltspire Avenger has protection from red.
Avacyn, the Purifier's ability resolves, and she deals 3 damage to each other creature and each opponent. The damage to Giltspire Avenger is prevented.
The opponent activates Giltspire Avenger's ability targeting Avacyn, the Purifier. This is legal because Avacyn, the Purifier has dealt damage to you this turn.
Giltspire Avenger's ability resolves, and Avacyn is destroyed.
If the opponent instead has 5 or less life, that scenario plays out differently starting from step 4:
Faith's Shield resolves. The opponent and each permanent they control has protection from red.
Avacyn, the Purifier's ability resolves. All damage it would deal is prevented.
Avacyn has not dealt damage to the opponent, so it is an illegal target for Giltspire Avenger's ability.
Your opponent can do what he wants (pro-Red his Avenger and destroy your Avacyn).
Both Avacyn's flip and Avacyn's damage are triggered abilities that go on the stack. As such, they can be responded to since both players get priority before they resolve. (See this question about how the stack works)
There is a small kink in this scenario: if your opponent is at 5 or less life, they also get pro-Red, so no damage is dealt to him/her, and Avenger can't tap to destroy Avacyn (yet).
Is it possible for him to cast it for pro-red, before Avacyn's transform into Purifier deals it lethal damage?
Yes, players can cast instants in response to triggered abilities being added to the stack.
They can cast Faith's Shield in response to the triggering of Archangel Avacyn's delayed triggered ability.
They can cast Faith's Shield in response to the triggering of Avacyn, the Purifier's triggered ability.
Faith's Shield will resolve before the ability resolves, giving Protection from Red, preventing the damage from Purifier.
If so and even if not so, does the 3 damage Avacyn does to opponent allow Giltspire to tap and destroy it?
Yes. If you took damage from Purifier, Giltspire can be used to destroy it.
You'll have to use it on your opponent's turn (since that's when you took the damage), but that's not a problem since you may activate activated abilities on an opponent's turn.
