Can I react to a Quickdraw Equipment being played and Gust the target before the Might addition resolves?

To answer your question: No, you cannot react to the Gear card itself being played, but you can react to the "When you play this, attach it to a Unit you control" triggered ability.

If you use Gust to remove the targeted unit from the battlefield before that triggered ability resolves, the attachment will fail.

The Sequence of Events

1. Playing the Gear: Your opponent plays a Gear with Quick-Draw. The Gear enters play at their base immediately. This action does not use the chain, and you cannot react to the Gear card being played (RiftJudge FAQ #8442).
2. Triggered Ability: The Quick-Draw keyword creates a "When you play this, attach it to a Unit you control" triggered ability. This ability is placed on the chain and targets a unit.
3. Reaction Window: Because this triggered ability is now on the chain, you gain priority and may play Gust as a reaction.
4. Resolution:
   • If you play Gust targeting the unit the Gear is attempting to attach to, Gust resolves first (LIFO).
   • The unit is removed from the battlefield and returned to its owner's hand.
   • When the Quick-Draw triggered ability attempts to resolve, its target is no longer on the battlefield. Because the target is now invalid, the instruction to attach cannot be followed. The ability "whiffs" (resolves with no effect), and the Gear remains unattached at the opponent's base (RiftJudge FAQ #8442, #3886).

Important Clarifications

• Interrupting vs. Mistargeting: You are not "interrupting" the resolution of the ability itself; you are using a reaction to change the game state (removing the target) before the ability resolves. Once the ability begins to resolve, it checks for its target. If the target is gone, the instruction to attach is ignored (Rule 359.3.e.6).
• Might Addition: You mentioned "before the might addition resolves." Note that the Might bonus provided by an Equipment is a passive effect that applies only while the Equipment is attached (Rule 137.3.a). If the attachment fails, the Equipment never attaches, and therefore its Might bonus never applies.

This is my interpretation based on the provided rules and FAQs. No official FAQ exists for this specific interaction beyond the established mechanics of Quick-Draw and targeting.