Do the Greige get the rewards for units defeated by Toxic Secretion or Toxic Fumes?

Yes. When a Unit is defeated by a Greige-controlled Toxic Secretion Talent or by a Toxic Fumes chip, the Greige are responsible for the Defeat and get the rewards for it. Toxic Secretion from Landmark minions does not count as Greige defeats, as landmarks are not Greige Units. If a Defeat from Toxic Secretion occurs from Echri's Toxic Secretion she gained from the Queen's Chamber Fortress Advancement, Echri levels up. In the rare instance where there are multiple different Toxic Secretion talents and which one caused the Defeat matters, the player whose Turn it is decides the order in which the Damage is applied.