If you attack with your “Giant Rat” and it is destroyed as a result of battle, you may attack with the monster Special Summoned by “Giant Rat” before entering Main Phase 2.
If your opponent controls your “Giant Rat” and it is destroyed as a result of battle, you get the effect of “Giant Rat” NOT your opponent.
Since the Special Summoning occurs during the Damage Step, “Torrential Tribute”, etc. cannot be used.