No. Because Fiendish Chain targets a monster, Obelisk the Tormentor will not be affected by it.
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When the Monarch is tribute summoned, it has a triggered effect that activates. Fiendish Chain can be chained to this. Fiendish Chain will resolve first, meaning the Monarch effect will be negated when it tries to resolve.
Yes. Fiendish Chain would work on Azure-Eyes Silver Dragon on it's summoning effect. The moment Azure-Eyes is summoned you can chain link Fiendish Chain to Azure-Eyes's effect creating a chain link 2. It would resolve as chain link 2: Fiendish Chain, then chain link 1 Azure-Eyes effect.
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.
Yes. Obelisk's protections only prevent his summon from being negated, and stops him from being targeted. Counter Traps almost never actually target anything, Divine Punishment is no exception and can chain to Obelisk's effect activation to negate it and destroy him. This means it can be taken out by cards cards that choose by a factor, such as Fissure and Smashing Ground.
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.