Montage Dragon does not have any activated effects so it has nothing that can be chained to.
However if Fiendish Chain is used on it, then it will not be able to attack, and its continuous effect that grants it an ATK value will be negated, and it will be treated as 0.
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.
No. Because Fiendish Chain targets a monster, Obelisk the Tormentor will not be affected by it.
If a monster activates an effect, then Fiendish Chain can be chained to this. Fiendish Chain will resolve first, meaning the monster's effect will be negated when it tries to resolve. So if Chaos Sorcerer activates his Ignition Effect, Fiendish Chain can be used to negate it.
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.
no fiendish chain most certainly does not negate attacks, if the card does not state it negates an attack it does not negate attacks
Yes. Once Red-Eyes Darkness Metal Dragon is successfully summoned, there is a summon response window for cards like Bottomless Trap Hole. Only Spell Speed 2 or 3 effects can be used here. You can therefore use Fiendish Chain against it, which will resolve if not negated. That means that when Red-Eyes Darkness Metal Dragon is able to activate its Ignition Effect, it will be negated when it tries to resolve. However it is worth noting that you could always wait until Red-Eyes Darkness Metal Dragon tries to activate the effect, and chain Fiendish Chain to it, it will resolve first and therefore negate the dragon's effect too. So hitting it before it can activate is not actually necessary.
The turn player has Priority, so can activate Chaos Sorcerer's effect as chain link 1. Fiendish Chain can be chained to this. Fiendish Chain will resolve first, so Chaos Sorcerer's effect will be negated when it tries to resolve.
Fiendish chain stays targeted to the monster and negates its effect when you enter end phase
Yes. Effect Veiler can chain to their activations in the main phase, and Fiendish Chain can chain to them at any time outside the damage step (meaning if they are attacked and flipped, it can't negate them).
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.
No. As soon as you enter the end phase, Fiendish will effect the monster.