Does first strike damage resolve on the stack or is it handled outside the stack?

Combat damage — including first strike damage — is never placed on the stack. It is assigned and dealt as a turn-based action governed by the rules of the combat phase, specifically during the Combat Damage Step (CR 510). Turn-based actions happen automatically and do not use the stack (CR 117.3b).

What does use the stack is the Combat Damage Step itself being divided into two steps when a creature with first strike or double strike is involved. After first strike damage is dealt, players receive priority and may cast spells or activate abilities before the regular combat damage step begins (CR 510.5). So players get a window to respond to damage results (e.g., a creature dying), but the damage itself was never on the stack.

When combat damage is assigned, each attacking or blocking creature with first strike deals its damage in the first combat damage step (CR 510.1). Creatures without first strike deal damage in the second combat damage step (CR 510.5). In both cases, damage is dealt simultaneously among all creatures in that step as a single event — again, not on the stack.

Example: Your 2/2 First Strike creature blocks an opponent's 3/3. In the first combat damage step, your 2/2 deals 2 damage to the 3/3 — not as a spell or ability on the stack, but as a game action. Players then get priority; if no one acts, the 3/3 is still alive (at 1 remaining toughness) and deals 3 damage in the regular combat damage step, killing your creature. Neither damage event was ever on the stack.

In summary: you cannot 'respond to' combat damage being dealt by countering it or removing it from the stack, because it was never there. You can only respond during the priority windows before or after each damage step.