“Your opponent’s Effect Monster” means the monster that your opponent controls when this effect is activated.
If “Ancient Gear Beast” attacks, the effects of Spell or Trap Cards that are already face-up can still have their effects activated, like “Ultimate Offering”, “Royal Oppression”, or “Skull Lair”. However, face-down Spell or Trap Cards cannot be activated (flipped face-up), including a face-down “Ultimate Offering”, etc.
The effect of “Ancient Gear Beast” that prevents your opponent from activating Spell or Trap Cards is a Continuous Effect. It cannot be chained to. “Divine Wrath” cannot be activated against it.
If “Ancient Gear Beast” attacks, and is destroyed before the end of the Damage Step, its effect is no longer applied after it is destroyed, even if it is still on the field and has not yet been sent to the Graveyard. (A Continuous Effect is no longer applied after the monster generating it has been destroyed, even if the monster is still on the field.) The effect of “Ancient Gear Beast” is applied up until the point it is destroyed, though.