"Dora of Fate"'s effect only applies when you Summon a monster, not when your opponent Summons a monster.
"Dora of Fate" targets: the face-up monster on your opponent's side of the field.
If you Summon multiple times with "Ultimate Offering", "Dora of Fate" only applies to the first Summon.
You cannot use "Barrel Behind the Door" against "Dora of Fate".
You get the "Dora of Fate"'s effect even if the monster you selected (on your opponent's side of the field) is destroyed before you Summon.