It doesn't matter who started with Priority, the simple matter is, if a monster has an activated effect, then Fiendish Chain can be chained to it. Fiendish Chain will resolve first, and the monster effect will be negated before it can resolve.
The main problem is against monsters that tribute themselves as a cost such as Exiled Force or Stardust Dragon. Because they leave the field as the cost of their own activations, Fiendish Chain would not be able to target them.
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The following monsters inflict damage to your opponent when destroyed. Some of these effects will depend on whether you or your opponent destroyed it and/or if it was destroyed in battle or by a card effect:AbakiEarthbound Immortal Aslla piscuBalloon LizardMecha-Dog MarronOxygeddon
Magical hats can only be used during your opponent's battle phase. Seeing as they can't flip their monsters during their battle phase, no. However, if your opponent does activate a spell or trap card during the battle phase, you can use Magical hats and protect your monsters. Overall, yes. You can use magical hats to protect your monsters from targeting effects as long as it's during your opponent's battle phase.
Flip Effects are Triggered Effects, so if a player flipped a flip effect monster, the effect will trigger as chain link 1. Fiendish Chain can be chained to it, to negate the effect when it tries to resolve. However, if the monster is flipped by an attack, then the trigger happens during the Damage Step. There are restrictions against what you are allowed to activate during the damage step, and Fiendish Chain can't be used here. So it could not stop a Flip Effect at that time.
Yes, it protects all dragons, regardless of who controls them. None of them can be targetted by spells, traps, or monster effects.
If you are referring to Destiny Hero Plasma's ability that states, "You can target 1 monster your opponent controls; equip that target to this card...", then the answer is no, this effect cannot be activated on your opponent's turn. This is because the effect only has a Spell Speed of 1 (for more on "Spell Speed", see the Related Links section below).That being said, Plasma's other effect that negates the effects of your opponent's face-up monsters is continuous and still works during your opponent's turn.