If I play Brynhir Thundersong and my opponent reacts with Tornado on their Kharox, does the Kharox player still get to play a card from my trash?

Yes — the Kharox player still gets to play a unit from your trash. Kharox's "When I become [Empowered]" trigger, and the reflexive "choose a unit in their trash and play it" trigger it creates (Rule 388.1), are both placed on the chain above Brynhir's "opponents can't play cards this turn" trigger, so they resolve first (Rule 340.1, newest item first).

Walk it through: Brynhir enters and her trigger goes on the chain; the opponent reveals Tornado Warrior, whose "when you play me from face down" trigger sits above it and resolves first, Empowering Kharox; Kharox's trigger then resolves (they Burn 3) and spawns the reflexive play trigger above Brynhir's trigger, which resolves and plays the unit from your trash before Brynhir's restriction ever applies. Once Brynhir's trigger finally resolves, the lockout is live and they can't play anything else that turn.

So the play window is real but narrow — it exists only while Brynhir's trigger is still on the chain, exactly as FAQ #2688/#814 describe. If they pass instead of acting there, the unit in your trash becomes unplayable for the rest of the turn.