First Strike says that this creature deals combat damage before other creatures. Now that combat damage doesn't use the stack, when is the First Striking creature actually dealing the damage? Is there a way to respond to it?
When First or Double Strike is involved there is a completely separate combat damage step created where that damage is dealt. It remains impossible to 'respond to' damage, but both players will get priority and can cast spells/activate abilities in that step, after First Strike damage, but before regular damage.
There's a first strike damage step and a regular damage step. You can't "respond" to first strike damage (because, as you said, it doesn't use the stack), but you do get an opportunity to play spells and activate abilities before and between the two.
Creatures with First Strike or Double Strike deal their damage in the First Strike Combat Damage Step.
You cannot respond to the assignment/dealing of combat damage since it no longer uses the stack but there is an opportunity for players to play spells/activate abilities in the First Strike Damage step before moving the normal Combat Damage Step.
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Both players get priority before and after first strike damage. For example:
If my opponent attacks with an Ajani's Pridemate that has two +1/+1 counters on it, and is 4/4. I block with Vampire Hexmage. Vampire Hexmage deals 2 first strike damage to Ajani's Pridemate, and then both players get priority to cast spells, during which I can sacrifice the Hexmage to remove the counters from the Pridemate, which will result in the Pridemate dying from the two damage already on it.