If I conquer Zaun Warrens with Rumble Hotheaded and discard a Mech from hand with Zaun Warrens, can I play it with Rumble?

No. Rumble's conquer trigger requires a valid Mech already in your trash when the trigger is placed on the Chain — you can't use Zaun Warrens' discard to create a target retroactively.

Zaun Warrens' "discard 1, then draw 1" resolves after both triggers are already on the Chain (they resolve LIFO). By the time you discard the Mech from hand, Rumble's trigger was already finalized without a legal target, and "you may" only affects whether you execute the effect on resolution, not target legality at declaration (Rule 352.4.b; FAQ #401).

If you already had a different Mech in trash, that one is playable — just not the one you discard from hand this turn.