Does first strike damage go on the stack?

Combat damage does not use the stack. When it's time for combat damage to be dealt, players assign damage, then it's all dealt simultaneously—no player can respond between assignment and dealing (CR 509.2).

First strike (and double strike) works by creating an additional combat damage step. If any attacking or blocking creature has first strike or double strike, there is a first combat damage step where only those creatures deal damage, then a second step where normal (and double strike) creatures deal damage (CR 702.7b, 702.4b).

Each combat damage step follows the same rule: damage is assigned and dealt without using the stack (CR 509.2). Players do get priority after each combat damage step resolves, so you can cast spells between the two steps—but the damage itself never sits on the stack waiting to be responded to.