If I use Stare Down and target Kha'Zix, Mutating Horror, and it starts a showdown, will Kha'Zix get the attacker designation first, get +2 might, and then Stare Down will trigger for 6 might, or will it use the Kha'Zix that was targeted before it gets moved?

No — Stare Down never moves Kha'Zix or gives it the Attacker designation, so there's no version where Stare Down itself turns Kha'Zix into a 6-Might unit. Stare Down chooses a friendly unit and a battlefield, then moves enemy units with less Might than the chosen unit to their base; your Kha'Zix stays where it is. Attacker/Defender designations are only assigned when a Combat Showdown opens, so clearing the enemies out (which just stages a non-combat showdown in the following Cleanup if the battlefield is contested) does nothing for Kha'Zix's "when I attack or defend" trigger. Also note Stare Down has neither Action nor Reaction, so you can't even play it once a showdown or combat is in progress (Rule 308.1.a) — it goes in your Main Phase while the turn is Neutral.

The timing intuition you have is right for the part that matters though: Stare Down compares Might as it resolves, using the chosen unit's current Might. So if Kha'Zix already has the +2 this turn (from actually attacking or defending earlier that turn), Stare Down counts it as a 6-Might unit and sweeps enemy units up to 5 Might.